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MCQ MATHEMATICAL PROBLEMS ON INDEFINITE INTEGRALS WITH ANSWERS TEST YOUR KNOWLEDGE OF ANTIDERIVATIVES AND INTEGRATION TECHNIQUES

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MCQ MATHEMATICAL PROBLEMS ON INDEFINITE
INTEGRALS WITH ANSWERS: TEST YOUR KNOWLEDGE OF
ANTIDERIVATIVES AND INTEGRATION TECHNIQUES
1. ∫ x³ dx
a) x⁴/4 + C b) 3x⁴/4 + C c) x⁴ + C d) 3x² + C
Answer: a) x⁴/4 + C
2. ∫ sin(x) dx
a) -cos(x) + C b) cos(x) + C c) tan(x) + C d) sec(x) + C
Answer: a) -cos(x) + C
3. ∫ e^x dx
a) e^x + C b) ln|x| + C c) xe^x + C d) e^x/x + C
Answer: a) e^x + C
4. ∫ 1/x dx
a) x + C b) ln|x| + C c) 1/x + C d) x² + C
Answer: b) ln|x| + C
5. ∫ cos(x) dx
a) sin(x) + C b) -sin(x) + C c) tan(x) + C d) sec(x) + C
Answer: a) sin(x) + C
6. ∫ x^n dx (where n ≠ -1)
a) x^(n+1)/(n+1) + C b) nx^(n-1) + C c) x^n/n + C d) x^(n+1) + C
Answer: a) x^(n+1)/(n+1) + C
7. ∫ tan(x) dx
a) sin(x) + C b) cos(x) + C c) -ln|cos(x)| + C d) sec(x) + C
Answer: c) -ln|cos(x)| + C
8. ∫ sec(x) dx
a) tan(x) + C b) ln|sec(x) + tan(x)| + C c) ln|cos(x)| + C d) csc(x) + C
Answer: b) ln|sec(x) + tan(x)| + C
9. ∫ csc(x) dx
a) ln|csc(x) - cot(x)| + C b) -ln|cos(x)| + C c) tan(x) + C d) sec(x) + C
Answer: a) ln|csc(x) - cot(x)| + C
10. ∫ cot(x) dx
a) sin(x) + C b) cos(x) + C c) ln|sin(x)| + C d) tan(x) + C
Answer: c) ln|sin(x)| + C
11. ∫ sec²(x) dx
a) sec(x) + C b) tan(x) + C c) ln|sec(x)| + C d) csc(x) + C
Answer: b) tan(x) + C
12. ∫ csc²(x) dx
a) -cot(x) + C b) tan(x) + C c) sec(x) + C d) -csc(x) + C
Answer: a) -cot(x) + C
13. ∫ sec(x)tan(x) dx
a) tan(x) + C b) sec(x) + C c) ln|cos(x)| + C d) ln|sec(x)| + C
Answer: b) sec(x) + C
14. ∫ csc(x)cot(x) dx
a) csc(x) + C b) -csc(x) + C c) ln|sin(x)| + C d) ln|csc(x)| + C
Answer: b) -csc(x) + C
15. ∫ 1/(1+x²) dx
a) tan ¹(x) + C b) cot ¹(x) + C c) sec ¹(x) + C d) csc ¹(x) + C⁻ ⁻ ⁻ ⁻
Answer: a) tan ¹(x) + C⁻
16. ∫ 1/√(1-x²) dx
a) sin ¹(x) + C b) cos ¹(x) + C c) tan ¹(x) + C d) cot ¹(x) + C⁻ ⁻ ⁻ ⁻
Answer: a) sin ¹(x) + C⁻
17. ∫ 1/√(x²-1) dx
a) sin ¹(x) + C b) cos ¹(x) + C c) sec ¹(x) + C d) csc ¹(x) + C⁻ ⁻ ⁻ ⁻
Answer: c) sec ¹(x) + C⁻
18. ∫ 1/(x√(x²-1)) dx
a) tan ¹(x) + C b) cot ¹(x) + C c) sec ¹(x) + C d) csc ¹(x) + C⁻ ⁻ ⁻ ⁻
Answer: d) csc ¹(x) + C⁻
19. ∫ ln(x) dx
a) x ln(x) + C b) x ln(x) - x + C c) ln(x)/x + C d) x/ln(x) + C
Answer: b) x ln(x) - x + C
20. ∫ x ln(x) dx
a) x²ln(x)/2 + C b) x²ln(x)/2 - x²/4 + C c) x²/2ln(x) + C d) ln(x²)/2 + C
Answer: b) x²ln(x)/2 - x²/4 + C
21. ∫ e^(ax) dx
a) e^(ax) + C b) e^(ax)/a + C c) ae^(ax) + C d) ln(a)e^(ax) + C
Answer: b) e^(ax)/a + C
22. ∫ sin(ax) dx
a) cos(ax)/a + C b) -cos(ax)/a + C c) sin(ax)/a + C d) -sin(ax)/a + C
Answer: b) -cos(ax)/a + C
23. ∫ cos(ax) dx
a) sin(ax)/a + C b) -sin(ax)/a + C c) cos(ax)/a + C d) -cos(ax)/a + C
Answer: a) sin(ax)/a + C
24. ∫ tan(ax) dx
a) -ln|cos(ax)|/a + C b) ln|sec(ax)|/a + C c) cot(ax)/a + C d) sec(ax)/a + C
Answer: b) ln|sec(ax)|/a + C
25. ∫ sec(ax) dx
a) tan(ax)/a + C b) ln|sec(ax) + tan(ax)|/a + C c) ln|cos(ax)|/a + C d) csc(ax)/a + C
Answer: b) ln|sec(ax) + tan(ax)|/a + C
26. ∫ csc(ax) dx
a) ln|csc(ax) - cot(ax)|/a + C b) -ln|sin(ax)|/a + C c) cot(ax)/a + C d) sec(ax)/a + C
Answer: a) ln|csc(ax) - cot(ax)|/a + C
27. ∫ cot(ax) dx
a) sin(ax)/a + C b) cos(ax)/a + C c) ln|sin(ax)|/a + C d) tan(ax)/a + C
Answer: c) ln|sin(ax)|/a + C
28. ∫ sec²(ax) dx
a) sec(ax)/a + C b) tan(ax)/a + C c) ln|sec(ax)|/a + C d) csc(ax)/a + C
Answer: b) tan(ax)/a + C
29. ∫ csc²(ax) dx
a) -cot(ax)/a + C b) tan(ax)/a + C c) sec(ax)/a + C d) -csc(ax)/a + C
Answer: a) -cot(ax)/a + C
30. ∫ sec(ax)tan(ax) dx
a) tan(ax)/a + C b) sec(ax)/a + C c) ln|cos(ax)|/a + C d) ln|sec(ax)|/a + C
Answer: b) sec(ax)/a + C
31. ∫ csc(ax)cot(ax) dx
a) csc(ax)/a + C b) -csc(ax)/a + C c) ln|sin(ax)|/a + C d) ln|csc(ax)|/a + C
Answer: b) -csc(ax)/a + C
32. ∫ 1/(a²+x²) dx
a) tan ¹(x/a)/a + C b) cot ¹(x/a)/a + C c) sec ¹(x/a)/a + C d) csc ¹(x/a)/a + C⁻ ⁻ ⁻ ⁻
Answer: a) tan ¹(x/a)/a + C⁻
33. ∫ 1/√(a²-x²) dx
a) sin ¹(x/a) + C b) cos ¹(x/a) + C c) tan ¹(x/a) + C d) cot ¹(x/a) + C⁻ ⁻ ⁻ ⁻
Answer: a) sin ¹(x/a) + C⁻
34. ∫ 1/√(x²-a²) dx
a) sin ¹(x/a) + C b) cos ¹(x/a) + C c) sec ¹(x/a) + C d) csc ¹(x/a) + C⁻ ⁻ ⁻ ⁻
Answer: c) sec ¹(x/a) + C⁻
35. ∫ 1/(x√(x²-a²)) dx
a) tan ¹(x/a) + C b) cot ¹(x/a) + C c) sec ¹(x/a) + C d) csc ¹(x/a) + C⁻ ⁻ ⁻ ⁻
Answer: d) csc ¹(x/a) + C⁻
36. ∫ x^n e^x dx
a) x^n e^x + C b) x^(n+1) e^x/(n+1) + C c) e^x (x^n - nx^(n-1) + n(n-1)x^(n-2) - ...) + C d) x^n
ln|x| + C
Answer: c) e^x (x^n - nx^(n-1) + n(n-1)x^(n-2) - ...) + C
37. ∫ x sin(x) dx
a) -x cos(x) + sin(x) + C b) x sin(x) - cos(x) + C c) x cos(x) + sin(x) + C d) -x sin(x) + cos(x) + C
Answer: a) -x cos(x) + sin(x) + C
38. ∫ x cos(x) dx
a) x sin(x) + cos(x) + C b) x cos(x) - sin(x) + C c) -x sin(x) + cos(x) + C d) x sin(x) - cos(x) + C
Answer: a) x sin(x) + cos(x) + C
39. ∫ x² sin(x) dx
a) -x² cos(x) + 2x sin(x) + 2 cos(x) + C b) x² sin(x) - 2x cos(x) - 2 sin(x) + C
c) -x² cos(x) + 2x sin(x) - 2 cos(x) + C d) x² cos(x) + 2x sin(x) + 2 cos(x) + C
Answer: a) -x² cos(x) + 2x sin(x) + 2 cos(x) + C
40. ∫ x² cos(x) dx
a) x² sin(x) - 2x cos(x) + 2 sin(x) + C b) x² cos(x) + 2x sin(x) - 2 cos(x) + C
c) -x² sin(x) + 2x cos(x) + 2 sin(x) + C d) x² sin(x) + 2x cos(x) - 2 sin(x) + C
Answer: a) x² sin(x) - 2x cos(x) + 2 sin(x) + C
41. ∫ e^x sin(x) dx
a) e^x (sin(x) - cos(x))/2 + C b) e^x (sin(x) + cos(x))/2 + C
c) e^x sin(x) + C d) e^x cos(x) + C
Answer: a) e^x (sin(x) - cos(x))/2 + C
42. ∫ e^x cos(x) dx
a) e^x (sin(x) + cos(x))/2 + C b) e^x (cos(x) - sin(x))/2 + C
c) e^x sin(x) + C d) e^x cos(x) + C
Answer: a) e^x (sin(x) + cos(x))/2 + C
43. ∫ 1/(x ln(x)) dx
a) ln|ln(x)| + C b) ln|x| + C c) 1/ln(x) + C d) x/ln(x) + C
Answer: a) ln|ln(x)| + C
44. ∫ x^2 e^(x^3) dx
a) e^(x^3)/3 + C b) x e^(x^3) + C c) e^(x^3)/(3x) + C d) x^3 e^(x^3) + C
Answer: a) e^(x^3)/3 + C
45. ∫ sin²(x) dx
a) x/2 - sin(2x)/4 + C b) x/2 + sin(2x)/4 + C c) x - sin(2x)/2 + C d) sin(x) + C
Answer: a) x/2 - sin(2x)/4 + C
46. ∫ cos²(x) dx
a) x/2 + sin(2x)/4 + C b) x/2 - sin(2x)/4 + C c) x + sin(2x)/2 + C d) cos(x) + C
Answer: a) x/2 + sin(2x)/4 + C
47. ∫ tan²(x) dx
a) tan(x) - x + C b) tan(x) + x + C c) -cot(x) - x + C d) cot(x) - x + C
Answer: a) tan(x) - x + C
48. ∫ sec²(x) tan(x) dx
a) sec²(x)/2 + C b) tan²(x)/2 + C c) sec(x) tan(x) + C d) sec(x) + C
Answer: a) sec²(x)/2 + C
49. ∫ csc²(x) cot(x) dx
a) -csc²(x)/2 + C b) cot²(x)/2 + C c) -csc(x) cot(x) + C d) csc(x) + C
Answer: a) -csc²(x)/2 + C
50. ∫ 1/(x²+a²) dx
a) tan ¹(x/a)/a + C b) cot ¹(x/a)/a + C c) (1/a) tan ¹(x/a) + C d) a tan ¹(x/a) + C⁻ ⁻ ⁻ ⁻
Answer: c) (1/a) tan ¹(x/a) + C⁻
51. ∫ 1/(x²-a²) dx
a) (1/2a) ln|(x-a)/(x+a)| + C b) (1/a) ln|x-a| + C c) (1/2a) ln|x²-a²| + C d) (1/a) ln|x+a| + C
Answer: a) (1/2a) ln|(x-a)/(x+a)| + C
52. ∫ 1/(a²-x²) dx
a) (1/2a) ln|(a+x)/(a-x)| + C b) (1/a) ln|a-x| + C c) (1/2a) ln|a²-x²| + C d) (1/a) ln|a+x| + C
Answer: a) (1/2a) ln|(a+x)/(a-x)| + C
53. ∫ x/(x²+a²) dx
a) ln|x²+a²|/2 + C b) a tan ¹(x/a) + C c) ln|x+a| + C d) x tan ¹(x/a) + C⁻ ⁻
Answer: a) ln|x²+a²|/2 + C
54. ∫ x/(x²-a²) dx
a) ln|x²-a²|/2 + C b) a coth ¹(x/a) + C c) ln|x-a| + C d) x coth ¹(x/a) + C⁻ ⁻
Answer: a) ln|x²-a²|/2 + C
55. ∫ x/(a²-x²) dx
a) -ln|a²-x²|/2 + C b) a tanh ¹(x/a) + C c) ln|a-x| + C d) x tanh ¹(x/a) + C⁻ ⁻
Answer: a) -ln|a²-x²|/2 + C
56. ∫ √(x²+a²) dx
a) x√(x²+a²)/2 + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) ln|x+√(x²+a²)|
+ C
Answer: a) x√(x²+a²)/2 + (a²/2)ln|x+√(x²+a²)| + C
57. ∫ √(x²-a²) dx
a) x√(x²-a²)/2 - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) ln|x+√(x²-a²)| + C
Answer: a) x√(x²-a²)/2 - (a²/2)ln|x+√(x²-a²)| + C
58. ∫ √(a²-x²) dx
a) x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²-x²)^(3/2)/3 + C d) arcsin(x/a) + C
Answer: a) x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
59. ∫ 1/(x√(x²-a²)) dx
a) arcsin(a/x) + C b) arcsec(x/a) + C c) -arccsc(x/a) + C d) arccos(a/x) + C
Answer: c) -arccsc(x/a) + C
60. ∫ 1/(x√(a²-x²)) dx
a) -arcsec(x/a) + C b) arcsin(x/a) + C c) arccsc(x/a) + C d) arccos(x/a) + C
Answer: c) arccsc(x/a) + C
61. ∫ x^n ln(x) dx
a) x^(n+1)ln(x)/(n+1) - x^(n+1)/(n+1)² + C b) x^n ln(x) + C c) x^(n+1)ln(x) + C d)
ln(x^(n+1))/(n+1) + C
Answer: a) x^(n+1)ln(x)/(n+1) - x^(n+1)/(n+1)² + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
62. ∫ (ln(x))² dx
a) x(ln(x))² - 2x ln(x) + 2x + C b) x(ln(x))² + C c) (ln(x))³/3 + C d) x²(ln(x))² + C
Answer: a) x(ln(x))² - 2x ln(x) + 2x + C
63. ∫ x^n e^(ax) dx
a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx b) x^n e^(ax) + C c) e^(ax) x^n/n + C d) a x^n e^(ax) +
C
Answer: a) x^n e^(ax)/a - (n/a) ∫ x^(n-1) e^(ax) dx
64. ∫ sin(ln(x)) dx
a) x sin(ln(x))/2 - x cos(ln(x))/2 + C b) -x cos(ln(x)) + C c) x sin(ln(x)) + C d) -sin(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 - x cos(ln(x))/2 + C
65. ∫ cos(ln(x)) dx
a) x sin(ln(x))/2 + x cos(ln(x))/2 + C b) x sin(ln(x)) + C c) x cos(ln(x)) + C d) cos(x)/ln(x) + C
Answer: a) x sin(ln(x))/2 + x cos(ln(x))/2 + C
66. ∫ x^n sin(x) dx
a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx b) x^n sin(x) + C c) sin(x^(n+1))/(n+1) + C d) -
cos(x^(n+1))/(n+1) + C
Answer: a) -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
67. ∫ x^n cos(x) dx
a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx b) x^n cos(x) + C c) cos(x^(n+1))/(n+1) + C d)
sin(x^(n+1))/(n+1) + C
Answer: a) x^n sin(x) - n ∫ x^(n-1) sin(x) dx
68. ∫ e^(ax) sin(bx) dx
a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C b) e^(ax) sin(bx) / a + C c) e^(ax) cos(bx) / b + C d)
e^(ax) (sin(bx) + cos(bx)) / (a + b) + C
Answer: a) e^(ax) (a sin(bx) - b cos(bx)) / (a² + b²) + C
69. ∫ e^(ax) cos(bx) dx
a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C b) e^(ax) cos(bx) / a + C c) e^(ax) sin(bx) / b + C
d) e^(ax) (cos(bx) - sin(bx)) / (a - b) + C
Answer: a) e^(ax) (a cos(bx) + b sin(bx)) / (a² + b²) + C
70. ∫ 1 / (a + b cos(x)) dx
a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C b) (1/b) tan ¹((a tan(x/2))/b) + C⁻ ⁻
c) (1/a) ln|a + b cos(x)| + C d) (1/√(a²-b²)) ln|(a+b)/(a-b)| + C
Answer: a) (2/√(a²-b²)) tan ¹((√(a²-b²) tan(x/2))/(a+b)) + C (when a > b)⁻
71. ∫ 1 / (a + b sin(x)) dx
a) (1/√(a²-b²)) ln|(a+b sin(x)+√(a²-b²))/(a+b sin(x)-√(a²-b²))| + C b) (1/b) ln|a + b sin(x)| + C
c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C d) (1/a) tan ¹((b cos(x))/a) + C⁻ ⁻
Answer: c) (2/√(a²-b²)) tan ¹((√(a²-b²) cot(x/2))/(a-b)) + C (when a > b)⁻
72. ∫ 1 / (x²√(x²-a²)) dx
a) -√(x²-a²)/(a²x) + C b) ln|x+√(x²-a²)|/a² + C c) -1/(a²√(x²-a²)) + C d) tan ¹(x/√(x²-a²))/a² + C⁻
Answer: a) -√(x²-a²)/(a²x) + C
73. ∫ x / √(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x arcsin(x/a) + C d) a arcsin(x/a) + C
Answer: a) -√(a²-x²) + C
74. ∫ x² / √(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x² arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
75. ∫ x³ / √(a²-x²) dx
a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C b) x²√(a²-x²) + C c) (a²/4)√(a²-x²) + C d) x³
arcsin(x/a) + C
Answer: a) -(x²/3 + a²/2)√(a²-x²) + (a⁴/3)arcsin(x/a) + C
76. ∫ √(x²+a²) / x dx
a) √(x²+a²) + a ln|x+√(x²+a²)| + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + a ln|x+√(x²+a²)| + C
77. ∫ √(x²-a²) / x dx
a) √(x²-a²) + a ln|x+√(x²-a²)| + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + a ln|x+√(x²-a²)| + C
78. ∫ x / √(x²+a²) dx
a) √(x²+a²) + C b) ln|x+√(x²+a²)| + C c) arcsinh(x/a) + C d) x√(x²+a²) + C
Answer: a) √(x²+a²) + C
79. ∫ x / √(x²-a²) dx
a) √(x²-a²) + C b) ln|x+√(x²-a²)| + C c) arccosh(x/a) + C d) x√(x²-a²) + C
Answer: a) √(x²-a²) + C
80. ∫ dx / (x√(x²-a²))
a) (1/a) ln|(√(x²-a²)-a)/x| + C b) (1/a) arcsec(x/a) + C c) (1/a) arccos(a/x) + C d) (1/a) ln|x+√(x²-a²)|
+ C
Answer: b) (1/a) arcsec(x/a) + C
81. ∫ dx / (x√(a²-x²))
a) -(1/a) ln|(√(a²-x²)+a)/x| + C b) (1/a) arccsc(x/a) + C c) (1/a) arcsin(a/x) + C d) (1/a) ln|x+√(a²-
x²)| + C
Answer: b) (1/a) arccsc(x/a) + C
82. ∫ x²√(a²-x²) dx
a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C b) x³√(a²-x²)/3 + C c) a²x√(a²-x²)/2 + C d)
x²arcsin(x/a) + C
Answer: a) (x/8)(2x²-a²)√(a²-x²) + (a⁴/8)arcsin(x/a) + C
83. ∫ x³√(a²-x²) dx
a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C b) x⁴√(a²-x²)/4 + C c) a²x²√(a²-x²)/3 + C d) x³arcsin(x/a) + C
Answer: a) -(a²-x²)^(3/2)(3a²+2x²)/15 + C
84. ∫ dx / (x²√(a²-x²))
a) -√(a²-x²)/(ax) + C b) arcsin(x/a)/a² + C c) ln|x+√(a²-x²)|/a² + C d) -1/(a√(a²-x²)) + C
Answer: a) -√(a²-x²)/(ax) + C
85. ∫ x dx / (a²+x²)²
a) -1/(2(a²+x²)) + C b) x/(2a²(a²+x²)) + C c) arctan(x/a)/(2a³) + C d) ln(a²+x²)/(2a²) + C
Answer: b) x/(2a²(a²+x²)) + C
86. ∫ x² dx / (a²+x²)²
a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C b) -1/(2(a²+x²)) + C c) x³/(3a²(a²+x²)) + C d) ln(a²+x²)/(2a) +
C
Answer: a) x/(2(a²+x²)) - arctan(x/a)/(2a) + C
87. ∫ dx / (x√(x²-a²))
a) arccos(a/x)/a + C b) arcsec(x/a)/a + C c) ln|x+√(x²-a²)|/a + C d) √(x²-a²)/(ax) + C
Answer: b) arcsec(x/a)/a + C
88. ∫ x dx / √(a²-x²)³
a) -1/√(a²-x²) + C b) arcsin(x/a)/a + C c) x/√(a²-x²) + C d) -√(a²-x²)/x + C
Answer: a) -1/√(a²-x²) + C
89. ∫ x² dx / √(a²-x²)³
a) -x/√(a²-x²) + arcsin(x/a) + C b) -√(a²-x²)/x + C c) x²/√(a²-x²) + C d) -a²/√(a²-x²) + C
Answer: a) -x/√(a²-x²) + arcsin(x/a) + C
90. ∫ dx / (x²√(x²+a²))
a) -√(x²+a²)/(ax²) + C b) arctan(x/a)/a² + C c) ln|x+√(x²+a²)|/a² + C d) -1/(a√(x²+a²)) + C
Answer: a) -√(x²+a²)/(ax²) + C
91. ∫ x dx / (a²-x²)²
a) -1/(2(a²-x²)) + C b) -x/(2a²(a²-x²)) + C c) arctanh(x/a)/(2a³) + C d) ln|a²-x²|/(2a²) + C
Answer: b) -x/(2a²(a²-x²)) + C
92. ∫ x² dx / (a²-x²)²
a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C b) 1/(2(a²-x²)) + C c) -x³/(3a²(a²-x²)) + C d) ln|a²-x²|/(2a) + C
Answer: a) x/(2(x²-a²)) + arctanh(x/a)/(2a) + C
93. ∫ √(x²+a²) dx
a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C b) x√(x²+a²) + C c) (x²+a²)^(3/2)/3 + C d) arcsinh(x/a)
+ C
Answer: a) (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
94. ∫ √(x²-a²) dx
a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C b) x√(x²-a²) + C c) (x²-a²)^(3/2)/3 + C d) arccosh(x/a) +
C
Answer: a) (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C
95. ∫ dx / (x²-a²)
a) (1/2a)ln|(x-a)/(x+a)| + C b) arctan(x/a)/a + C c) ln|x²-a²|/2a + C d) 1/(ax) + C
Answer: a) (1/2a)ln|(x-a)/(x+a)| + C
96. ∫ dx / (x²+a²)²
a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C b) 1/(a²+x²) + C c) -1/(2a²√(x²+a²)) + C d) ln(x²+a²)/(2a²)
+ C
Answer: a) x/(2a²(x²+a²)) + arctan(x/a)/(2a³) + C
97. ∫ x√(a²-x²) dx
a) -(a²-x²)^(3/2)/3 + C b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C c) (a²/2)√(a²-x²) + C d) x²√(a²-x²)/2 +
C
Answer: b) -x√(a²-x²)/2 + (a²/2)arcsin(x/a) + C
98. ∫ x/√(a²-x²) dx
a) -√(a²-x²) + C b) arcsin(x/a) + C c) x√(a²-x²) + C d) ln|x+√(a²-x²)| + C
Answer: a) -√(a²-x²) + C
99. ∫ x²/√(a²-x²) dx
a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C b) x√(a²-x²) + C c) (a²/3)√(a²-x²) + C d) x²arcsin(x/a) + C
Answer: a) (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C
100. ∫ dx / (x√(x²+a²))
a) (1/a)ln|(√(x²+a²)-a)/x| + C b) (1/a)arcsec(x/a) + C c) (1/a)arcsinh(x/a) + C d) (1/a)ln|x+√(x²+a²)|
+ C
Answer: c) (1/a)arcsinh(x/a) + C
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