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MCQ MATHEMATICAL PROBLEMS WITH ANSWERS ON INTEGRATION BY SUBSTITUTIONMULTIPLE CHOICE QUESTIONS INTEGRATION BY SUBSTITUTION

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MCQ MATHEMATICAL PROBLEMS WITH ANSWERS ON
INTEGRATION BY SUBSTITUTION:MULTIPLE CHOICE
QUESTIONS: INTEGRATION BY SUBSTITUTION
1. ∫ cos(3x) dx
a) 1/3 sin(3x) + C
b) -1/3 sin(3x) + C
c) sin(3x) + C
d) cos(3x) + C
Answer: a) 1/3 sin(3x) + C
2. ∫ e^(2x) dx
a) 1/2 e^(2x) + C
b) e^(2x) + C
c) 2e^(2x) + C
d) e^x + C
Answer: a) 1/2 e^(2x) + C
3. ∫ (1/x) dx
a) x + C
b) ln|x| + C
c) 1/x + C
d) x^2/2 + C
Answer: b) ln|x| + C
4. ∫ sin(5x) dx
a) -1/5 cos(5x) + C
b) 1/5 cos(5x) + C
c) sin(5x) + C
d) 5 cos(5x) + C
Answer: a) -1/5 cos(5x) + C
5. ∫ (1/√x) dx
a) 2√x + C
b) x^(3/2)/3 + C
c) 2x^(3/2)/3 + C
d) 2√x + C
Answer: c) 2x^(3/2)/3 + C
6. ∫ tan(x) dx
a) sec(x) + C
b) -ln|cos(x)| + C
c) sin(x) + C
d) cot(x) + C
Answer: b) -ln|cos(x)| + C
7. ∫ sec^2(x) dx
a) tan(x) + C
b) sec(x) + C
c) -cot(x) + C
d) ln|sec(x) + tan(x)| + C
Answer: a) tan(x) + C
8. ∫ x e^x dx
a) e^x + C
b) x e^x - e^x + C
c) x e^x + C
d) e^x (x - 1) + C
Answer: b) x e^x - e^x + C
9. ∫ 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
10. ∫ x^2 e^x dx
a) x^2 e^x - 2x e^x + 2e^x + C
b) e^x (x^2 - 2x + 2) + C
c) x^2 e^x + C
d) e^x (x^2 + 2x + 2) + C
Answer: b) e^x (x^2 - 2x + 2) + C
11. ∫ sin^2(x) dx
a) x/2 - sin(2x)/4 + C
b) -cos(2x)/4 + C
c) sin(2x)/2 + C
d) x - sin(2x)/2 + C
Answer: a) x/2 - sin(2x)/4 + C
12. ∫ cos^2(x) dx
a) x/2 + sin(2x)/4 + C
b) sin(2x)/2 + C
c) cos(2x)/4 + C
d) x + sin(2x)/2 + C
Answer: a) x/2 + sin(2x)/4 + C
13. ∫ sec(x) tan(x) dx
a) tan(x) + C
b) sec(x) + C
c) ln|sec(x)| + C
d) csc(x) + C
Answer: b) sec(x) + C
14. ∫ csc(x) cot(x) dx
a) csc(x) + C
b) -csc(x) + C
c) ln|csc(x)| + C
d) sec(x) + C
Answer: b) -csc(x) + C
15. ∫ x^3 dx
a) x^4/3 + C
b) 3x^4/4 + C
c) x^4/4 + C
d) 4x^3/3 + C
Answer: c) x^4/4 + C
16. ∫ 1/(1+x^2) dx
a) tan^(-1)(x) + C
b) cot^(-1)(x) + C
c) sec^(-1)(x) + C
d) sin^(-1)(x) + C
Answer: a) tan^(-1)(x) + C
17. ∫ 1/√(1-x^2) dx
a) tan^(-1)(x) + C
b) cos^(-1)(x) + C
c) sin^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: c) sin^(-1)(x) + C
18. ∫ 1/(x√(x^2-1)) dx
a) tan^(-1)(x) + C
b) cos^(-1)(x) + C
c) sin^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
19. ∫ x^n dx (n ≠ -1)
a) x^(n+1)/(n+1) + C
b) x^n/n + C
c) nx^(n-1) + C
d) x^(n+1) + C
Answer: a) x^(n+1)/(n+1) + C
20. ∫ cos(ln(x)) dx
a) x sin(ln(x)) + C
b) x/2 (sin(ln(x)) + cos(ln(x))) + C
c) x cos(ln(x)) + C
d) sin(ln(x)) + C
Answer: b) x/2 (sin(ln(x)) + cos(ln(x))) + C
21. ∫ sin(ln(x)) dx
a) x cos(ln(x)) + C
b) x/2 (cos(ln(x)) - sin(ln(x))) + C
c) x sin(ln(x)) + C
d) cos(ln(x)) + C
Answer: b) x/2 (cos(ln(x)) - sin(ln(x))) + C
22. ∫ 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
23. ∫ x^2 ln(x) dx
a) x^3 ln(x)/3 - x^3/9 + C
b) x^3 ln(x)/3 - x^2/2 + C
c) x^2 ln(x) - 2x + C
d) x^3 ln(x) - x^3/3 + C
Answer: a) x^3 ln(x)/3 - x^3/9 + C
24. ∫ e^(x^2) x dx
a) e^(x^2)/2 + C
b) x e^(x^2) + C
c) e^(x^2)/(2x) + C
d) x^2 e^(x^2) + C
Answer: a) e^(x^2)/2 + C
25. ∫ x sin(x^2) dx
a) -1/2 cos(x^2) + C
b) sin(x^2)/2 + C
c) x cos(x^2) + C
d) -cos(x^2)/(2x) + C
Answer: a) -1/2 cos(x^2) + C
26. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) sin(x^3)/3 + C
c) x cos(x^3) + C
d) -cos(x^3)/(3x) + C
Answer: a) -1/3 cos(x^3) + C
27. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) cos(x^2)/2 + C
c) x sin(x^2) + C
d) sin(x^2)/(2x) + C
Answer: a) 1/2 sin(x^2) + C
28. ∫ tan^2(x) dx
a) tan(x) - x + C
b) -cot(x) - x + C
c) sec^2(x)/2 + C
d) tan(x) + C
Answer: a) tan(x) - x + C
29. ∫ cot^2(x) dx
a) -tan(x) + x + C
b) cot(x) - x + C
c) -csc^2(x)/2 + C
d) -cot(x) + C
Answer: b) cot(x) - x + C
30. ∫ sec^3(x) dx
a) tan(x) sec(x)/2 + C
b) sec(x) tan(x)/2 + ln|sec(x) + tan(x)| + C
c) sec(x) + C
d) tan(x) + C
Answer: b) sec(x) tan(x)/2 + ln|sec(x) + tan(x)| + C
31. ∫ csc^3(x) dx
a) -cot(x) csc(x)/2 + C
b) -csc(x) cot(x)/2 - ln|csc(x) + cot(x)| + C
c) csc(x) + C
d) -cot(x) + C
Answer: b) -csc(x) cot(x)/2 - ln|csc(x) + cot(x)| + C
32. ∫ x^2 cos(x) dx
a) x^2 sin(x) + 2x cos(x) - 2 sin(x) + C
b) x^2 sin(x) - 2x cos(x) + 2 sin(x) + C
c) x sin(x) - cos(x) + C
d) x^2 cos(x) - 2x sin(x) + C
Answer: b) x^2 sin(x) - 2x cos(x) + 2 sin(x) + C
33. ∫ x sin(x) dx
a) -x cos(x) + sin(x) + C
b) x sin(x) - cos(x) + C
c) sin(x) - x cos(x) + C
d) x cos(x) + sin(x) + C
Answer: a) -x cos(x) + sin(x) + C
34. ∫ 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
35. ∫ x ln(x^2) dx
a) x^2 ln(x^2)/2 - x^2/2 + C
b) x^2 ln(x)/2 - x^2/4 + C
c) x ln(x^2) - 2x + C
d) x^2 ln(x^2) - x^2 + C
Answer: b) x^2 ln(x)/2 - x^2/4 + C
36. ∫ sin^3(x) dx
a) -cos(x) + cos^3(x)/3 + C
b) cos(x) - cos^3(x)/3 + C
c) sin(x) - sin^3(x)/3 + C
d) -cos(x) (1 - 2sin^2(x))/3 + C
Answer: a) -cos(x) + cos^3(x)/3 + C
37. ∫ cos^3(x) dx
a) sin(x) - sin^3(x)/3 + C
b) sin(x) (2 + cos^2(x))/3 + C
c) cos(x) + cos^3(x)/3 + C
d) sin(x) (1 - 2cos^2(x))/3 + C
Answer: b) sin(x) (2 + cos^2(x))/3 + C
38. ∫ 1/(x^2 + 1)^2 dx
a) x/(2(x^2 + 1)) + tan^(-1)(x)/2 + C
b) x/(x^2 + 1) + C
c) 1/(x^2 + 1) + C
d) tan^(-1)(x) + C
Answer: a) x/(2(x^2 + 1)) + tan^(-1)(x)/2 + C
39. ∫ x^2 / (x^2 + 1) dx
a) x - tan^(-1)(x) + C
b) x + tan^(-1)(x) + C
c) ln(x^2 + 1)/2 + C
d) x - ln(x^2 + 1)/2 + C
Answer: a) x - tan^(-1)(x) + C
40. ∫ x / (x^2 + 1) dx
a) tan^(-1)(x) + C
b) ln(x^2 + 1)/2 + C
c) x/(x^2 + 1) + C
d) 1/(x^2 + 1) + C
Answer: b) ln(x^2 + 1)/2 + C
41. ∫ x^3 / (x^2 + 1) dx
a) (x^2 - 1)/2 + tan^(-1)(x) + C
b) x^2/2 - ln(x^2 + 1)/2 + C
c) (x^2 + 1)/2 - tan^(-1)(x) + C
d) x^2/2 + ln(x^2 + 1)/2 + C
Answer: b) x^2/2 - ln(x^2 + 1)/2 + C
42. ∫ x / √(1 - x^2) dx
a) -√(1 - x^2) + C
b) sin^(-1)(x) + C
c) cos^(-1)(x) + C
d) tan^(-1)(x) + C
Answer: a) -√(1 - x^2) + C
43. ∫ 1 / (x^2 √(1 - x^2)) dx
a) -√(1 - x^2) / x + C
b) √(1 - x^2) / x + C
c) -1 / (x√(1 - x^2)) + C
d) 1 / (x√(1 - x^2)) + C
Answer: c) -1 / (x√(1 - x^2)) + C
44. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
c) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: b) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
45. ∫ √(1 - x^2) dx
a) (x√(1 - x^2) + sin^(-1)(x))/2 + C
b) (x√(1 - x^2) - sin^(-1)(x))/2 + C
c) x√(1 - x^2)/2 + sin^(-1)(x) + C
d) x√(1 - x^2)/2 - sin^(-1)(x) + C
Answer: a) (x√(1 - x^2) + sin^(-1)(x))/2 + C
46. ∫ 1 / (x^2 + 4) dx
a) 1/2 tan^(-1)(x/2) + C
b) 1/4 tan^(-1)(x/2) + C
c) 1/2 tan^(-1)(2x) + C
d) 1/4 tan^(-1)(2x) + C
Answer: a) 1/2 tan^(-1)(x/2) + C
47. ∫ 1 / (x^2 - 4) dx
a) 1/4 ln|(x-2)/(x+2)| + C
b) 1/2 ln|(x-2)/(x+2)| + C
c) 1/4 ln|x^2 - 4| + C
d) 1/2 ln|x^2 - 4| + C
Answer: a) 1/4 ln|(x-2)/(x+2)| + C
48. ∫ 1 / (x^2 + 2x + 2) dx
a) tan^(-1)(x+1) + C
b) 1/2 tan^(-1)(x+1) + C
c) ln(x^2 + 2x + 2) + C
d) 1/2 ln(x^2 + 2x + 2) + C
Answer: b) 1/2 tan^(-1)(x+1) + C
49. ∫ x / (x^2 + 4) dx
a) 1/2 ln(x^2 + 4) + C
b) 1/4 ln(x^2 + 4) + C
c) tan^(-1)(x/2) + C
d) 1/2 tan^(-1)(x/2) + C
Answer: b) 1/4 ln(x^2 + 4) + C
50. ∫ x / (x^2 - 4) dx
a) 1/2 ln|x^2 - 4| + C
b) 1/4 ln|x^2 - 4| + C
c) 1/2 ln|(x-2)/(x+2)| + C
d) 1/4 ln|(x-2)/(x+2)| + C
Answer: a) 1/2 ln|x^2 - 4| + C
51. ∫ x^3 / (x^2 + 1)^2 dx
a) x^2 / (2(x^2 + 1)) + 1/2 tan^(-1)(x) + C
b) x^2 / (2(x^2 + 1)) - 1/2 tan^(-1)(x) + C
c) x / (x^2 + 1) + tan^(-1)(x) + C
d) x / (x^2 + 1) - tan^(-1)(x) + C
Answer: b) x^2 / (2(x^2 + 1)) - 1/2 tan^(-1)(x) + C
52. ∫ e^(2x) sin(3x) dx
a) e^(2x) (2sin(3x) - 3cos(3x)) / 13 + C
b) e^(2x) (3sin(3x) - 2cos(3x)) / 13 + C
c) e^(2x) (2sin(3x) + 3cos(3x)) / 13 + C
d) e^(2x) (3sin(3x) + 2cos(3x)) / 13 + C
Answer: a) e^(2x) (2sin(3x) - 3cos(3x)) / 13 + C
53. ∫ x / √(x^2 + 1) dx
a) √(x^2 + 1) + C
b) ln(x + √(x^2 + 1)) + C
c) sinh^(-1)(x) + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 + 1) + C
54. ∫ 1 / (x √(x^2 + 1)) dx
a) ln(x + √(x^2 + 1)) + C
b) sinh^(-1)(x) + C
c) cosh^(-1)(x) + C
d) ln|x + √(x^2 + 1)| + C
Answer: d) ln|x + √(x^2 + 1)| + C
55. ∫ x^2 / √(x^2 + 1) dx
a) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
Answer: c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
56. ∫ √(x^2 + 1) dx
a) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
Answer: b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
57. ∫ 1 / (x^2 √(x^2 + 1)) dx
a) -√(x^2 + 1) / x + C
b) √(x^2 + 1) / x + C
c) -1 / (x√(x^2 + 1)) + C
d) 1 / (x√(x^2 + 1)) + C
Answer: c) -1 / (x√(x^2 + 1)) + C
58. ∫ x / (x^2 + 4)^2 dx
a) -1 / (4(x^2 + 4)) + C
b) 1 / (4(x^2 + 4)) + C
c) -x / (2(x^2 + 4)) + C
d) x / (2(x^2 + 4)) + C
Answer: a) -1 / (4(x^2 + 4)) + C
59. ∫ x^2 / (x^2 + 4)^2 dx
a) x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
c) -x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
d) -x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
Answer: b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
60. ∫ x^3 / (x^2 + 4)^2 dx
a) (x^2 - 4) / (4(x^2 + 4)) + C
b) (x^2 + 4) / (4(x^2 + 4)) + C
c) (x^2 - 4) / (2(x^2 + 4)) + C
d) (x^2 + 4) / (2(x^2 + 4)) + C
Answer: a) (x^2 - 4) / (4(x^2 + 4)) + C
61. ∫ sin(ln(x)) / x dx
a) -cos(ln(x))/2 + sin(ln(x))/2 + C
b) cos(ln(x))/2 - sin(ln(x))/2 + C
c) -cos(ln(x))/2 - sin(ln(x))/2 + C
d) cos(ln(x))/2 + sin(ln(x))/2 + C
Answer: a) -cos(ln(x))/2 + sin(ln(x))/2 + C
62. ∫ cos(ln(x)) / x dx
a) sin(ln(x))/2 + cos(ln(x))/2 + C
b) sin(ln(x))/2 - cos(ln(x))/2 + C
c) -sin(ln(x))/2 + cos(ln(x))/2 + C
d) -sin(ln(x))/2 - cos(ln(x))/2 + C
Answer: a) sin(ln(x))/2 + cos(ln(x))/2 + C
63. ∫ tan(ln(x)) / x dx
a) -ln|cos(ln(x))| + C
b) ln|cos(ln(x))| + C
c) ln|sin(ln(x))| + C
d) -ln|sin(ln(x))| + C
Answer: b) ln|cos(ln(x))| + C
64. ∫ sec(ln(x)) / x dx
a) tan(ln(x)) + C
b) -tan(ln(x)) + C
c) cot(ln(x)) + C
d) -cot(ln(x)) + C
Answer: a) tan(ln(x)) + C
65. ∫ csc(ln(x)) / x dx
a) ln|csc(ln(x)) - cot(ln(x))| + C
b) -ln|csc(ln(x)) - cot(ln(x))| + C
c) ln|tan(ln(x)/2)| + C
d) -ln|tan(ln(x)/2)| + C
Answer: d) -ln|tan(ln(x)/2)| + C
66. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) -cos(x^3)/3 + C
c) sin(x^3)/3 + C
d) cos(x^3)/3 + C
Answer: b) -cos(x^3)/3 + C
67. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) -1/2 sin(x^2) + C
c) cos(x^2)/2 + C
d) -cos(x^2)/2 + C
Answer: a) 1/2 sin(x^2) + C
68. ∫ e^(sin(x)) cos(x) dx
a) e^(sin(x)) + C
b) -e^(sin(x)) + C
c) e^(-sin(x)) + C
d) -e^(-sin(x)) + C
Answer: a) e^(sin(x)) + C
69. ∫ x / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: d) 1/4 tan^(-1)(x^2) + C
70. ∫ x^3 / (x^4 + 1) dx
a) 1/2 ln(x^4 + 1) + C
b) 1/4 ln(x^4 + 1) + C
c) 1/2 tan^(-1)(x^2) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: b) 1/4 ln(x^4 + 1) + C
71. ∫ x^2 / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
72. ∫ 1 / (x^3 + x) dx
a) 1/2 ln|x^2 - x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
c) 1/2 ln|x^2 - x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/2 ln|x^2 + x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
73. ∫ x^2 / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) ln|x + 1| - (1/√3) tan^(-1)((2x - 1)/√3) + C
c) ln|x + 1| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|x^3 - 1| + C
Answer: a) 1/3 ln|x^3 + 1| + C
74. ∫ x / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
c) 1/3 ln|(x + 1)/(x^2 - x + 1)| - (2/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|(x - 1)/(x^2 + x + 1)| + (2/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
75. ∫ 1 / (x^2 (x^2 + 1)) dx
a) -1/x + tan^(-1)(x) + C
b) 1/x - tan^(-1)(x) + C
c) -1/x - tan^(-1)(x) + C
d) 1/x + tan^(-1)(x) + C
Answer: c) -1/x - tan^(-1)(x) + C
76. ∫ x^3 / (x^4 + 1) dx
a) 1/4 ln(x^4 + 1) + C
b) 1/2 ln(x^4 + 1) + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln(x^4 + 1) + C
77. ∫ x^2 / (x^4 - 1) dx
a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
b) 1/2 ln|(x^2 + 1)/(x^2 - 1)| + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
78. ∫ x / (x^4 - 1) dx
a) 1/4 ln|(x - 1)/(x + 1)| + C
b) 1/2 ln|(x - 1)/(x + 1)| + C
c) 1/4 ln|(x^2 - 1)| + C
d) 1/2 ln|(x^2 - 1)| + C
Answer: a) 1/4 ln|(x - 1)/(x + 1)| + C
79. ∫ 1 / (x^2 √(x^2 - 1)) dx
a) -√(x^2 - 1) / x + C
b) √(x^2 - 1) / x + C
c) -1 / (x√(x^2 - 1)) + C
d) 1 / (x√(x^2 - 1)) + C
Answer: c) -1 / (x√(x^2 - 1)) + C
80. ∫ x^2 / √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
81. ∫ √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
82. ∫ x / √(x^2 - 1) dx
a) √(x^2 - 1) + C
b) -√(x^2 - 1) + C
c) ln|x + √(x^2 - 1)| + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 - 1) + C
83. ∫ 1 / (x √(x^2 - 1)) dx
a) ln|x + √(x^2 - 1)| + C
b) -ln|x + √(x^2 - 1)| + C
c) cosh^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
84. ∫ x^3 / √(x^2 - 1) dx
a) (x^2 + 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
b) (x^2 - 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
d) (x^2 + 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
Answer: c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
85. ∫ 1 / (x^2 √(1 - x^2)) dx
a) √(1 - x^2) / x + C
b) -√(1 - x^2) / x + C
c) 1 / (x√(1 - x^2)) + C
d) -1 / (x√(1 - x^2)) + C
Answer: d) -1 / (x√(1 - x^2)) + C
86. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
87. ∫ x / √(1 - x^2) dx
a) √(1 - x^2) + C
b) -√(1 - x^2) + C
c) sin^(-1)(x) + C
d) -sin^(-1)(x) + C
Answer: b) -√(1 - x^2) + C
88. ∫ 1 / (x √(1 - x^2)) dx
a) ln|x + √(1 - x^2)| + C
b) -ln|x + √(1 - x^2)| + C
c) sin^(-1)(x) + C
d) cos^(-1)(x) + C
Answer: b) -ln|x + √(1 - x^2)| + C
89. ∫ x^3 / √(1 - x^2) dx
a) -(1 - x^2)^(3/2)/3 + C
b) (1 - x^2)^(3/2)/3 + C
c) -(1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
d) (1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
Answer: a) -(1 - x^2)^(3/2)/3 + C
90. ∫ 1 / (x^2 + x + 1) dx
a) 2/√3 tan^(-1)((2x + 1)/√3) + C
b) 2/√3 tan^(-1)((x + 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x + 1)/√3) + C
d) 1/√3 tan^(-1)((x + 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x + 1)/√3) + C
. ∫ √(x^2 + 1) dx
a) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
Answer: b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
57. ∫ 1 / (x^2 √(x^2 + 1)) dx
a) -√(x^2 + 1) / x + C
b) √(x^2 + 1) / x + C
c) -1 / (x√(x^2 + 1)) + C
d) 1 / (x√(x^2 + 1)) + C
Answer: c) -1 / (x√(x^2 + 1)) + C
58. ∫ x / (x^2 + 4)^2 dx
a) -1 / (4(x^2 + 4)) + C
b) 1 / (4(x^2 + 4)) + C
c) -x / (2(x^2 + 4)) + C
d) x / (2(x^2 + 4)) + C
Answer: a) -1 / (4(x^2 + 4)) + C
59. ∫ x^2 / (x^2 + 4)^2 dx
a) x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
c) -x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
d) -x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
Answer: b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
60. ∫ x^3 / (x^2 + 4)^2 dx
a) (x^2 - 4) / (4(x^2 + 4)) + C
b) (x^2 + 4) / (4(x^2 + 4)) + C
c) (x^2 - 4) / (2(x^2 + 4)) + C
d) (x^2 + 4) / (2(x^2 + 4)) + C
Answer: a) (x^2 - 4) / (4(x^2 + 4)) + C
61. ∫ sin(ln(x)) / x dx
a) -cos(ln(x))/2 + sin(ln(x))/2 + C
b) cos(ln(x))/2 - sin(ln(x))/2 + C
c) -cos(ln(x))/2 - sin(ln(x))/2 + C
d) cos(ln(x))/2 + sin(ln(x))/2 + C
Answer: a) -cos(ln(x))/2 + sin(ln(x))/2 + C
62. ∫ cos(ln(x)) / x dx
a) sin(ln(x))/2 + cos(ln(x))/2 + C
b) sin(ln(x))/2 - cos(ln(x))/2 + C
c) -sin(ln(x))/2 + cos(ln(x))/2 + C
d) -sin(ln(x))/2 - cos(ln(x))/2 + C
Answer: a) sin(ln(x))/2 + cos(ln(x))/2 + C
63. ∫ tan(ln(x)) / x dx
a) -ln|cos(ln(x))| + C
b) ln|cos(ln(x))| + C
c) ln|sin(ln(x))| + C
d) -ln|sin(ln(x))| + C
Answer: b) ln|cos(ln(x))| + C
64. ∫ sec(ln(x)) / x dx
a) tan(ln(x)) + C
b) -tan(ln(x)) + C
c) cot(ln(x)) + C
d) -cot(ln(x)) + C
Answer: a) tan(ln(x)) + C
65. ∫ csc(ln(x)) / x dx
a) ln|csc(ln(x)) - cot(ln(x))| + C
b) -ln|csc(ln(x)) - cot(ln(x))| + C
c) ln|tan(ln(x)/2)| + C
d) -ln|tan(ln(x)/2)| + C
Answer: d) -ln|tan(ln(x)/2)| + C
66. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) -cos(x^3)/3 + C
c) sin(x^3)/3 + C
d) cos(x^3)/3 + C
Answer: b) -cos(x^3)/3 + C
67. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) -1/2 sin(x^2) + C
c) cos(x^2)/2 + C
d) -cos(x^2)/2 + C
Answer: a) 1/2 sin(x^2) + C
68. ∫ e^(sin(x)) cos(x) dx
a) e^(sin(x)) + C
b) -e^(sin(x)) + C
c) e^(-sin(x)) + C
d) -e^(-sin(x)) + C
Answer: a) e^(sin(x)) + C
69. ∫ x / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: d) 1/4 tan^(-1)(x^2) + C
70. ∫ x^3 / (x^4 + 1) dx
a) 1/2 ln(x^4 + 1) + C
b) 1/4 ln(x^4 + 1) + C
c) 1/2 tan^(-1)(x^2) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: b) 1/4 ln(x^4 + 1) + C
71. ∫ x^2 / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
72. ∫ 1 / (x^3 + x) dx
a) 1/2 ln|x^2 - x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
c) 1/2 ln|x^2 - x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/2 ln|x^2 + x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
73. ∫ x^2 / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) ln|x + 1| - (1/√3) tan^(-1)((2x - 1)/√3) + C
c) ln|x + 1| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|x^3 - 1| + C
Answer: a) 1/3 ln|x^3 + 1| + C
74. ∫ x / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
c) 1/3 ln|(x + 1)/(x^2 - x + 1)| - (2/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|(x - 1)/(x^2 + x + 1)| + (2/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
75. ∫ 1 / (x^2 (x^2 + 1)) dx
a) -1/x + tan^(-1)(x) + C
b) 1/x - tan^(-1)(x) + C
c) -1/x - tan^(-1)(x) + C
d) 1/x + tan^(-1)(x) + C
Answer: c) -1/x - tan^(-1)(x) + C
76. ∫ x^3 / (x^4 + 1) dx
a) 1/4 ln(x^4 + 1) + C
b) 1/2 ln(x^4 + 1) + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln(x^4 + 1) + C
77. ∫ x^2 / (x^4 - 1) dx
a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
b) 1/2 ln|(x^2 + 1)/(x^2 - 1)| + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
78. ∫ x / (x^4 - 1) dx
a) 1/4 ln|(x - 1)/(x + 1)| + C
b) 1/2 ln|(x - 1)/(x + 1)| + C
c) 1/4 ln|(x^2 - 1)| + C
d) 1/2 ln|(x^2 - 1)| + C
Answer: a) 1/4 ln|(x - 1)/(x + 1)| + C
79. ∫ 1 / (x^2 √(x^2 - 1)) dx
a) -√(x^2 - 1) / x + C
b) √(x^2 - 1) / x + C
c) -1 / (x√(x^2 - 1)) + C
d) 1 / (x√(x^2 - 1)) + C
Answer: c) -1 / (x√(x^2 - 1)) + C
80. ∫ x^2 / √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
81. ∫ √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
82. ∫ x / √(x^2 - 1) dx
a) √(x^2 - 1) + C
b) -√(x^2 - 1) + C
c) ln|x + √(x^2 - 1)| + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 - 1) + C
83. ∫ 1 / (x √(x^2 - 1)) dx
a) ln|x + √(x^2 - 1)| + C
b) -ln|x + √(x^2 - 1)| + C
c) cosh^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
84. ∫ x^3 / √(x^2 - 1) dx
a) (x^2 + 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
b) (x^2 - 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
d) (x^2 + 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
Answer: c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
85. ∫ 1 / (x^2 √(1 - x^2)) dx
a) √(1 - x^2) / x + C
b) -√(1 - x^2) / x + C
c) 1 / (x√(1 - x^2)) + C
d) -1 / (x√(1 - x^2)) + C
Answer: d) -1 / (x√(1 - x^2)) + C
86. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
87. ∫ x / √(1 - x^2) dx
a) √(1 - x^2) + C
b) -√(1 - x^2) + C
c) sin^(-1)(x) + C
d) -sin^(-1)(x) + C
Answer: b) -√(1 - x^2) + C
88. ∫ 1 / (x √(1 - x^2)) dx
a) ln|x + √(1 - x^2)| + C
b) -ln|x + √(1 - x^2)| + C
c) sin^(-1)(x) + C
d) cos^(-1)(x) + C
Answer: b) -ln|x + √(1 - x^2)| + C
89. ∫ x^3 / √(1 - x^2) dx
a) -(1 - x^2)^(3/2)/3 + C
b) (1 - x^2)^(3/2)/3 + C
c) -(1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
d) (1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
Answer: a) -(1 - x^2)^(3/2)/3 + C
90. ∫ 1 / (x^2 + x + 1) dx
a) 2/√3 tan^(-1)((2x + 1)/√3) + C
b) 2/√3 tan^(-1)((x + 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x + 1)/√3) + C
d) 1/√3 tan^(-1)((x + 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x + 1)/√3) + C
. ∫ √(x^2 + 1) dx
a) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
Answer: b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
57. ∫ 1 / (x^2 √(x^2 + 1)) dx
a) -√(x^2 + 1) / x + C
b) √(x^2 + 1) / x + C
c) -1 / (x√(x^2 + 1)) + C
d) 1 / (x√(x^2 + 1)) + C
Answer: c) -1 / (x√(x^2 + 1)) + C
58. ∫ x / (x^2 + 4)^2 dx
a) -1 / (4(x^2 + 4)) + C
b) 1 / (4(x^2 + 4)) + C
c) -x / (2(x^2 + 4)) + C
d) x / (2(x^2 + 4)) + C
Answer: a) -1 / (4(x^2 + 4)) + C
59. ∫ x^2 / (x^2 + 4)^2 dx
a) x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
c) -x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
d) -x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
Answer: b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
60. ∫ x^3 / (x^2 + 4)^2 dx
a) (x^2 - 4) / (4(x^2 + 4)) + C
b) (x^2 + 4) / (4(x^2 + 4)) + C
c) (x^2 - 4) / (2(x^2 + 4)) + C
d) (x^2 + 4) / (2(x^2 + 4)) + C
Answer: a) (x^2 - 4) / (4(x^2 + 4)) + C
61. ∫ sin(ln(x)) / x dx
a) -cos(ln(x))/2 + sin(ln(x))/2 + C
b) cos(ln(x))/2 - sin(ln(x))/2 + C
c) -cos(ln(x))/2 - sin(ln(x))/2 + C
d) cos(ln(x))/2 + sin(ln(x))/2 + C
Answer: a) -cos(ln(x))/2 + sin(ln(x))/2 + C
62. ∫ cos(ln(x)) / x dx
a) sin(ln(x))/2 + cos(ln(x))/2 + C
b) sin(ln(x))/2 - cos(ln(x))/2 + C
c) -sin(ln(x))/2 + cos(ln(x))/2 + C
d) -sin(ln(x))/2 - cos(ln(x))/2 + C
Answer: a) sin(ln(x))/2 + cos(ln(x))/2 + C
63. ∫ tan(ln(x)) / x dx
a) -ln|cos(ln(x))| + C
b) ln|cos(ln(x))| + C
c) ln|sin(ln(x))| + C
d) -ln|sin(ln(x))| + C
Answer: b) ln|cos(ln(x))| + C
64. ∫ sec(ln(x)) / x dx
a) tan(ln(x)) + C
b) -tan(ln(x)) + C
c) cot(ln(x)) + C
d) -cot(ln(x)) + C
Answer: a) tan(ln(x)) + C
65. ∫ csc(ln(x)) / x dx
a) ln|csc(ln(x)) - cot(ln(x))| + C
b) -ln|csc(ln(x)) - cot(ln(x))| + C
c) ln|tan(ln(x)/2)| + C
d) -ln|tan(ln(x)/2)| + C
Answer: d) -ln|tan(ln(x)/2)| + C
66. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) -cos(x^3)/3 + C
c) sin(x^3)/3 + C
d) cos(x^3)/3 + C
Answer: b) -cos(x^3)/3 + C
67. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) -1/2 sin(x^2) + C
c) cos(x^2)/2 + C
d) -cos(x^2)/2 + C
Answer: a) 1/2 sin(x^2) + C
68. ∫ e^(sin(x)) cos(x) dx
a) e^(sin(x)) + C
b) -e^(sin(x)) + C
c) e^(-sin(x)) + C
d) -e^(-sin(x)) + C
Answer: a) e^(sin(x)) + C
69. ∫ x / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: d) 1/4 tan^(-1)(x^2) + C
70. ∫ x^3 / (x^4 + 1) dx
a) 1/2 ln(x^4 + 1) + C
b) 1/4 ln(x^4 + 1) + C
c) 1/2 tan^(-1)(x^2) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: b) 1/4 ln(x^4 + 1) + C
71. ∫ x^2 / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
72. ∫ 1 / (x^3 + x) dx
a) 1/2 ln|x^2 - x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
c) 1/2 ln|x^2 - x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/2 ln|x^2 + x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
73. ∫ x^2 / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) ln|x + 1| - (1/√3) tan^(-1)((2x - 1)/√3) + C
c) ln|x + 1| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|x^3 - 1| + C
Answer: a) 1/3 ln|x^3 + 1| + C
74. ∫ x / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
c) 1/3 ln|(x + 1)/(x^2 - x + 1)| - (2/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|(x - 1)/(x^2 + x + 1)| + (2/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
75. ∫ 1 / (x^2 (x^2 + 1)) dx
a) -1/x + tan^(-1)(x) + C
b) 1/x - tan^(-1)(x) + C
c) -1/x - tan^(-1)(x) + C
d) 1/x + tan^(-1)(x) + C
Answer: c) -1/x - tan^(-1)(x) + C
76. ∫ x^3 / (x^4 + 1) dx
a) 1/4 ln(x^4 + 1) + C
b) 1/2 ln(x^4 + 1) + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln(x^4 + 1) + C
77. ∫ x^2 / (x^4 - 1) dx
a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
b) 1/2 ln|(x^2 + 1)/(x^2 - 1)| + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
78. ∫ x / (x^4 - 1) dx
a) 1/4 ln|(x - 1)/(x + 1)| + C
b) 1/2 ln|(x - 1)/(x + 1)| + C
c) 1/4 ln|(x^2 - 1)| + C
d) 1/2 ln|(x^2 - 1)| + C
Answer: a) 1/4 ln|(x - 1)/(x + 1)| + C
79. ∫ 1 / (x^2 √(x^2 - 1)) dx
a) -√(x^2 - 1) / x + C
b) √(x^2 - 1) / x + C
c) -1 / (x√(x^2 - 1)) + C
d) 1 / (x√(x^2 - 1)) + C
Answer: c) -1 / (x√(x^2 - 1)) + C
80. ∫ x^2 / √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
81. ∫ √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
82. ∫ x / √(x^2 - 1) dx
a) √(x^2 - 1) + C
b) -√(x^2 - 1) + C
c) ln|x + √(x^2 - 1)| + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 - 1) + C
83. ∫ 1 / (x √(x^2 - 1)) dx
a) ln|x + √(x^2 - 1)| + C
b) -ln|x + √(x^2 - 1)| + C
c) cosh^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
84. ∫ x^3 / √(x^2 - 1) dx
a) (x^2 + 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
b) (x^2 - 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
d) (x^2 + 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
Answer: c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
85. ∫ 1 / (x^2 √(1 - x^2)) dx
a) √(1 - x^2) / x + C
b) -√(1 - x^2) / x + C
c) 1 / (x√(1 - x^2)) + C
d) -1 / (x√(1 - x^2)) + C
Answer: d) -1 / (x√(1 - x^2)) + C
86. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
87. ∫ x / √(1 - x^2) dx
a) √(1 - x^2) + C
b) -√(1 - x^2) + C
c) sin^(-1)(x) + C
d) -sin^(-1)(x) + C
Answer: b) -√(1 - x^2) + C
88. ∫ 1 / (x √(1 - x^2)) dx
a) ln|x + √(1 - x^2)| + C
b) -ln|x + √(1 - x^2)| + C
c) sin^(-1)(x) + C
d) cos^(-1)(x) + C
Answer: b) -ln|x + √(1 - x^2)| + C
89. ∫ x^3 / √(1 - x^2) dx
a) -(1 - x^2)^(3/2)/3 + C
b) (1 - x^2)^(3/2)/3 + C
c) -(1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
d) (1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
Answer: a) -(1 - x^2)^(3/2)/3 + C
90. ∫ 1 / (x^2 + x + 1) dx
a) 2/√3 tan^(-1)((2x + 1)/√3) + C
b) 2/√3 tan^(-1)((x + 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x + 1)/√3) + C
d) 1/√3 tan^(-1)((x + 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x + 1)/√3) + C
. ∫ √(x^2 + 1) dx
a) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
Answer: b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
57. ∫ 1 / (x^2 √(x^2 + 1)) dx
a) -√(x^2 + 1) / x + C
b) √(x^2 + 1) / x + C
c) -1 / (x√(x^2 + 1)) + C
d) 1 / (x√(x^2 + 1)) + C
Answer: c) -1 / (x√(x^2 + 1)) + C
58. ∫ x / (x^2 + 4)^2 dx
a) -1 / (4(x^2 + 4)) + C
b) 1 / (4(x^2 + 4)) + C
c) -x / (2(x^2 + 4)) + C
d) x / (2(x^2 + 4)) + C
Answer: a) -1 / (4(x^2 + 4)) + C
59. ∫ x^2 / (x^2 + 4)^2 dx
a) x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
c) -x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
d) -x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
Answer: b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
60. ∫ x^3 / (x^2 + 4)^2 dx
a) (x^2 - 4) / (4(x^2 + 4)) + C
b) (x^2 + 4) / (4(x^2 + 4)) + C
c) (x^2 - 4) / (2(x^2 + 4)) + C
d) (x^2 + 4) / (2(x^2 + 4)) + C
Answer: a) (x^2 - 4) / (4(x^2 + 4)) + C
61. ∫ sin(ln(x)) / x dx
a) -cos(ln(x))/2 + sin(ln(x))/2 + C
b) cos(ln(x))/2 - sin(ln(x))/2 + C
c) -cos(ln(x))/2 - sin(ln(x))/2 + C
d) cos(ln(x))/2 + sin(ln(x))/2 + C
Answer: a) -cos(ln(x))/2 + sin(ln(x))/2 + C
62. ∫ cos(ln(x)) / x dx
a) sin(ln(x))/2 + cos(ln(x))/2 + C
b) sin(ln(x))/2 - cos(ln(x))/2 + C
c) -sin(ln(x))/2 + cos(ln(x))/2 + C
d) -sin(ln(x))/2 - cos(ln(x))/2 + C
Answer: a) sin(ln(x))/2 + cos(ln(x))/2 + C
63. ∫ tan(ln(x)) / x dx
a) -ln|cos(ln(x))| + C
b) ln|cos(ln(x))| + C
c) ln|sin(ln(x))| + C
d) -ln|sin(ln(x))| + C
Answer: b) ln|cos(ln(x))| + C
64. ∫ sec(ln(x)) / x dx
a) tan(ln(x)) + C
b) -tan(ln(x)) + C
c) cot(ln(x)) + C
d) -cot(ln(x)) + C
Answer: a) tan(ln(x)) + C
65. ∫ csc(ln(x)) / x dx
a) ln|csc(ln(x)) - cot(ln(x))| + C
b) -ln|csc(ln(x)) - cot(ln(x))| + C
c) ln|tan(ln(x)/2)| + C
d) -ln|tan(ln(x)/2)| + C
Answer: d) -ln|tan(ln(x)/2)| + C
66. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) -cos(x^3)/3 + C
c) sin(x^3)/3 + C
d) cos(x^3)/3 + C
Answer: b) -cos(x^3)/3 + C
67. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) -1/2 sin(x^2) + C
c) cos(x^2)/2 + C
d) -cos(x^2)/2 + C
Answer: a) 1/2 sin(x^2) + C
68. ∫ e^(sin(x)) cos(x) dx
a) e^(sin(x)) + C
b) -e^(sin(x)) + C
c) e^(-sin(x)) + C
d) -e^(-sin(x)) + C
Answer: a) e^(sin(x)) + C
69. ∫ x / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: d) 1/4 tan^(-1)(x^2) + C
70. ∫ x^3 / (x^4 + 1) dx
a) 1/2 ln(x^4 + 1) + C
b) 1/4 ln(x^4 + 1) + C
c) 1/2 tan^(-1)(x^2) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: b) 1/4 ln(x^4 + 1) + C
71. ∫ x^2 / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
72. ∫ 1 / (x^3 + x) dx
a) 1/2 ln|x^2 - x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
c) 1/2 ln|x^2 - x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/2 ln|x^2 + x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
73. ∫ x^2 / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) ln|x + 1| - (1/√3) tan^(-1)((2x - 1)/√3) + C
c) ln|x + 1| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|x^3 - 1| + C
Answer: a) 1/3 ln|x^3 + 1| + C
74. ∫ x / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
c) 1/3 ln|(x + 1)/(x^2 - x + 1)| - (2/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|(x - 1)/(x^2 + x + 1)| + (2/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
75. ∫ 1 / (x^2 (x^2 + 1)) dx
a) -1/x + tan^(-1)(x) + C
b) 1/x - tan^(-1)(x) + C
c) -1/x - tan^(-1)(x) + C
d) 1/x + tan^(-1)(x) + C
Answer: c) -1/x - tan^(-1)(x) + C
76. ∫ x^3 / (x^4 + 1) dx
a) 1/4 ln(x^4 + 1) + C
b) 1/2 ln(x^4 + 1) + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln(x^4 + 1) + C
77. ∫ x^2 / (x^4 - 1) dx
a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
b) 1/2 ln|(x^2 + 1)/(x^2 - 1)| + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
78. ∫ x / (x^4 - 1) dx
a) 1/4 ln|(x - 1)/(x + 1)| + C
b) 1/2 ln|(x - 1)/(x + 1)| + C
c) 1/4 ln|(x^2 - 1)| + C
d) 1/2 ln|(x^2 - 1)| + C
Answer: a) 1/4 ln|(x - 1)/(x + 1)| + C
79. ∫ 1 / (x^2 √(x^2 - 1)) dx
a) -√(x^2 - 1) / x + C
b) √(x^2 - 1) / x + C
c) -1 / (x√(x^2 - 1)) + C
d) 1 / (x√(x^2 - 1)) + C
Answer: c) -1 / (x√(x^2 - 1)) + C
80. ∫ x^2 / √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
81. ∫ √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
82. ∫ x / √(x^2 - 1) dx
a) √(x^2 - 1) + C
b) -√(x^2 - 1) + C
c) ln|x + √(x^2 - 1)| + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 - 1) + C
83. ∫ 1 / (x √(x^2 - 1)) dx
a) ln|x + √(x^2 - 1)| + C
b) -ln|x + √(x^2 - 1)| + C
c) cosh^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
84. ∫ x^3 / √(x^2 - 1) dx
a) (x^2 + 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
b) (x^2 - 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
d) (x^2 + 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
Answer: c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
85. ∫ 1 / (x^2 √(1 - x^2)) dx
a) √(1 - x^2) / x + C
b) -√(1 - x^2) / x + C
c) 1 / (x√(1 - x^2)) + C
d) -1 / (x√(1 - x^2)) + C
Answer: d) -1 / (x√(1 - x^2)) + C
86. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
87. ∫ x / √(1 - x^2) dx
a) √(1 - x^2) + C
b) -√(1 - x^2) + C
c) sin^(-1)(x) + C
d) -sin^(-1)(x) + C
Answer: b) -√(1 - x^2) + C
88. ∫ 1 / (x √(1 - x^2)) dx
a) ln|x + √(1 - x^2)| + C
b) -ln|x + √(1 - x^2)| + C
c) sin^(-1)(x) + C
d) cos^(-1)(x) + C
Answer: b) -ln|x + √(1 - x^2)| + C
89. ∫ x^3 / √(1 - x^2) dx
a) -(1 - x^2)^(3/2)/3 + C
b) (1 - x^2)^(3/2)/3 + C
c) -(1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
d) (1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
Answer: a) -(1 - x^2)^(3/2)/3 + C
90. ∫ 1 / (x^2 + x + 1) dx
a) 2/√3 tan^(-1)((2x + 1)/√3) + C
b) 2/√3 tan^(-1)((x + 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x + 1)/√3) + C
d) 1/√3 tan^(-1)((x + 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x + 1)/√3) + C
. ∫ √(x^2 + 1) dx
a) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
Answer: b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
57. ∫ 1 / (x^2 √(x^2 + 1)) dx
a) -√(x^2 + 1) / x + C
b) √(x^2 + 1) / x + C
c) -1 / (x√(x^2 + 1)) + C
d) 1 / (x√(x^2 + 1)) + C
Answer: c) -1 / (x√(x^2 + 1)) + C
58. ∫ x / (x^2 + 4)^2 dx
a) -1 / (4(x^2 + 4)) + C
b) 1 / (4(x^2 + 4)) + C
c) -x / (2(x^2 + 4)) + C
d) x / (2(x^2 + 4)) + C
Answer: a) -1 / (4(x^2 + 4)) + C
59. ∫ x^2 / (x^2 + 4)^2 dx
a) x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
c) -x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
d) -x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
Answer: b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
60. ∫ x^3 / (x^2 + 4)^2 dx
a) (x^2 - 4) / (4(x^2 + 4)) + C
b) (x^2 + 4) / (4(x^2 + 4)) + C
c) (x^2 - 4) / (2(x^2 + 4)) + C
d) (x^2 + 4) / (2(x^2 + 4)) + C
Answer: a) (x^2 - 4) / (4(x^2 + 4)) + C
61. ∫ sin(ln(x)) / x dx
a) -cos(ln(x))/2 + sin(ln(x))/2 + C
b) cos(ln(x))/2 - sin(ln(x))/2 + C
c) -cos(ln(x))/2 - sin(ln(x))/2 + C
d) cos(ln(x))/2 + sin(ln(x))/2 + C
Answer: a) -cos(ln(x))/2 + sin(ln(x))/2 + C
62. ∫ cos(ln(x)) / x dx
a) sin(ln(x))/2 + cos(ln(x))/2 + C
b) sin(ln(x))/2 - cos(ln(x))/2 + C
c) -sin(ln(x))/2 + cos(ln(x))/2 + C
d) -sin(ln(x))/2 - cos(ln(x))/2 + C
Answer: a) sin(ln(x))/2 + cos(ln(x))/2 + C
63. ∫ tan(ln(x)) / x dx
a) -ln|cos(ln(x))| + C
b) ln|cos(ln(x))| + C
c) ln|sin(ln(x))| + C
d) -ln|sin(ln(x))| + C
Answer: b) ln|cos(ln(x))| + C
64. ∫ sec(ln(x)) / x dx
a) tan(ln(x)) + C
b) -tan(ln(x)) + C
c) cot(ln(x)) + C
d) -cot(ln(x)) + C
Answer: a) tan(ln(x)) + C
65. ∫ csc(ln(x)) / x dx
a) ln|csc(ln(x)) - cot(ln(x))| + C
b) -ln|csc(ln(x)) - cot(ln(x))| + C
c) ln|tan(ln(x)/2)| + C
d) -ln|tan(ln(x)/2)| + C
Answer: d) -ln|tan(ln(x)/2)| + C
66. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) -cos(x^3)/3 + C
c) sin(x^3)/3 + C
d) cos(x^3)/3 + C
Answer: b) -cos(x^3)/3 + C
67. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) -1/2 sin(x^2) + C
c) cos(x^2)/2 + C
d) -cos(x^2)/2 + C
Answer: a) 1/2 sin(x^2) + C
68. ∫ e^(sin(x)) cos(x) dx
a) e^(sin(x)) + C
b) -e^(sin(x)) + C
c) e^(-sin(x)) + C
d) -e^(-sin(x)) + C
Answer: a) e^(sin(x)) + C
69. ∫ x / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: d) 1/4 tan^(-1)(x^2) + C
70. ∫ x^3 / (x^4 + 1) dx
a) 1/2 ln(x^4 + 1) + C
b) 1/4 ln(x^4 + 1) + C
c) 1/2 tan^(-1)(x^2) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: b) 1/4 ln(x^4 + 1) + C
71. ∫ x^2 / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
72. ∫ 1 / (x^3 + x) dx
a) 1/2 ln|x^2 - x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
c) 1/2 ln|x^2 - x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/2 ln|x^2 + x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
73. ∫ x^2 / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) ln|x + 1| - (1/√3) tan^(-1)((2x - 1)/√3) + C
c) ln|x + 1| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|x^3 - 1| + C
Answer: a) 1/3 ln|x^3 + 1| + C
74. ∫ x / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
c) 1/3 ln|(x + 1)/(x^2 - x + 1)| - (2/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|(x - 1)/(x^2 + x + 1)| + (2/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
75. ∫ 1 / (x^2 (x^2 + 1)) dx
a) -1/x + tan^(-1)(x) + C
b) 1/x - tan^(-1)(x) + C
c) -1/x - tan^(-1)(x) + C
d) 1/x + tan^(-1)(x) + C
Answer: c) -1/x - tan^(-1)(x) + C
76. ∫ x^3 / (x^4 + 1) dx
a) 1/4 ln(x^4 + 1) + C
b) 1/2 ln(x^4 + 1) + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln(x^4 + 1) + C
77. ∫ x^2 / (x^4 - 1) dx
a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
b) 1/2 ln|(x^2 + 1)/(x^2 - 1)| + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
78. ∫ x / (x^4 - 1) dx
a) 1/4 ln|(x - 1)/(x + 1)| + C
b) 1/2 ln|(x - 1)/(x + 1)| + C
c) 1/4 ln|(x^2 - 1)| + C
d) 1/2 ln|(x^2 - 1)| + C
Answer: a) 1/4 ln|(x - 1)/(x + 1)| + C
79. ∫ 1 / (x^2 √(x^2 - 1)) dx
a) -√(x^2 - 1) / x + C
b) √(x^2 - 1) / x + C
c) -1 / (x√(x^2 - 1)) + C
d) 1 / (x√(x^2 - 1)) + C
Answer: c) -1 / (x√(x^2 - 1)) + C
80. ∫ x^2 / √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
81. ∫ √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
82. ∫ x / √(x^2 - 1) dx
a) √(x^2 - 1) + C
b) -√(x^2 - 1) + C
c) ln|x + √(x^2 - 1)| + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 - 1) + C
83. ∫ 1 / (x √(x^2 - 1)) dx
a) ln|x + √(x^2 - 1)| + C
b) -ln|x + √(x^2 - 1)| + C
c) cosh^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
84. ∫ x^3 / √(x^2 - 1) dx
a) (x^2 + 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
b) (x^2 - 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
d) (x^2 + 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
Answer: c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
85. ∫ 1 / (x^2 √(1 - x^2)) dx
a) √(1 - x^2) / x + C
b) -√(1 - x^2) / x + C
c) 1 / (x√(1 - x^2)) + C
d) -1 / (x√(1 - x^2)) + C
Answer: d) -1 / (x√(1 - x^2)) + C
86. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
87. ∫ x / √(1 - x^2) dx
a) √(1 - x^2) + C
b) -√(1 - x^2) + C
c) sin^(-1)(x) + C
d) -sin^(-1)(x) + C
Answer: b) -√(1 - x^2) + C
88. ∫ 1 / (x √(1 - x^2)) dx
a) ln|x + √(1 - x^2)| + C
b) -ln|x + √(1 - x^2)| + C
c) sin^(-1)(x) + C
d) cos^(-1)(x) + C
Answer: b) -ln|x + √(1 - x^2)| + C
89. ∫ x^3 / √(1 - x^2) dx
a) -(1 - x^2)^(3/2)/3 + C
b) (1 - x^2)^(3/2)/3 + C
c) -(1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
d) (1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
Answer: a) -(1 - x^2)^(3/2)/3 + C
90. ∫ 1 / (x^2 + x + 1) dx
a) 2/√3 tan^(-1)((2x + 1)/√3) + C
b) 2/√3 tan^(-1)((x + 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x + 1)/√3) + C
d) 1/√3 tan^(-1)((x + 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x + 1)/√3) + C
. ∫ √(x^2 + 1) dx
a) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
Answer: b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
57. ∫ 1 / (x^2 √(x^2 + 1)) dx
a) -√(x^2 + 1) / x + C
b) √(x^2 + 1) / x + C
c) -1 / (x√(x^2 + 1)) + C
d) 1 / (x√(x^2 + 1)) + C
Answer: c) -1 / (x√(x^2 + 1)) + C
58. ∫ x / (x^2 + 4)^2 dx
a) -1 / (4(x^2 + 4)) + C
b) 1 / (4(x^2 + 4)) + C
c) -x / (2(x^2 + 4)) + C
d) x / (2(x^2 + 4)) + C
Answer: a) -1 / (4(x^2 + 4)) + C
59. ∫ x^2 / (x^2 + 4)^2 dx
a) x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
c) -x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
d) -x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
Answer: b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
60. ∫ x^3 / (x^2 + 4)^2 dx
a) (x^2 - 4) / (4(x^2 + 4)) + C
b) (x^2 + 4) / (4(x^2 + 4)) + C
c) (x^2 - 4) / (2(x^2 + 4)) + C
d) (x^2 + 4) / (2(x^2 + 4)) + C
Answer: a) (x^2 - 4) / (4(x^2 + 4)) + C
61. ∫ sin(ln(x)) / x dx
a) -cos(ln(x))/2 + sin(ln(x))/2 + C
b) cos(ln(x))/2 - sin(ln(x))/2 + C
c) -cos(ln(x))/2 - sin(ln(x))/2 + C
d) cos(ln(x))/2 + sin(ln(x))/2 + C
Answer: a) -cos(ln(x))/2 + sin(ln(x))/2 + C
62. ∫ cos(ln(x)) / x dx
a) sin(ln(x))/2 + cos(ln(x))/2 + C
b) sin(ln(x))/2 - cos(ln(x))/2 + C
c) -sin(ln(x))/2 + cos(ln(x))/2 + C
d) -sin(ln(x))/2 - cos(ln(x))/2 + C
Answer: a) sin(ln(x))/2 + cos(ln(x))/2 + C
63. ∫ tan(ln(x)) / x dx
a) -ln|cos(ln(x))| + C
b) ln|cos(ln(x))| + C
c) ln|sin(ln(x))| + C
d) -ln|sin(ln(x))| + C
Answer: b) ln|cos(ln(x))| + C
64. ∫ sec(ln(x)) / x dx
a) tan(ln(x)) + C
b) -tan(ln(x)) + C
c) cot(ln(x)) + C
d) -cot(ln(x)) + C
Answer: a) tan(ln(x)) + C
65. ∫ csc(ln(x)) / x dx
a) ln|csc(ln(x)) - cot(ln(x))| + C
b) -ln|csc(ln(x)) - cot(ln(x))| + C
c) ln|tan(ln(x)/2)| + C
d) -ln|tan(ln(x)/2)| + C
Answer: d) -ln|tan(ln(x)/2)| + C
66. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) -cos(x^3)/3 + C
c) sin(x^3)/3 + C
d) cos(x^3)/3 + C
Answer: b) -cos(x^3)/3 + C
67. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) -1/2 sin(x^2) + C
c) cos(x^2)/2 + C
d) -cos(x^2)/2 + C
Answer: a) 1/2 sin(x^2) + C
68. ∫ e^(sin(x)) cos(x) dx
a) e^(sin(x)) + C
b) -e^(sin(x)) + C
c) e^(-sin(x)) + C
d) -e^(-sin(x)) + C
Answer: a) e^(sin(x)) + C
69. ∫ x / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: d) 1/4 tan^(-1)(x^2) + C
70. ∫ x^3 / (x^4 + 1) dx
a) 1/2 ln(x^4 + 1) + C
b) 1/4 ln(x^4 + 1) + C
c) 1/2 tan^(-1)(x^2) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: b) 1/4 ln(x^4 + 1) + C
71. ∫ x^2 / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
72. ∫ 1 / (x^3 + x) dx
a) 1/2 ln|x^2 - x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
c) 1/2 ln|x^2 - x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/2 ln|x^2 + x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
73. ∫ x^2 / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) ln|x + 1| - (1/√3) tan^(-1)((2x - 1)/√3) + C
c) ln|x + 1| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|x^3 - 1| + C
Answer: a) 1/3 ln|x^3 + 1| + C
74. ∫ x / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
c) 1/3 ln|(x + 1)/(x^2 - x + 1)| - (2/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|(x - 1)/(x^2 + x + 1)| + (2/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
75. ∫ 1 / (x^2 (x^2 + 1)) dx
a) -1/x + tan^(-1)(x) + C
b) 1/x - tan^(-1)(x) + C
c) -1/x - tan^(-1)(x) + C
d) 1/x + tan^(-1)(x) + C
Answer: c) -1/x - tan^(-1)(x) + C
76. ∫ x^3 / (x^4 + 1) dx
a) 1/4 ln(x^4 + 1) + C
b) 1/2 ln(x^4 + 1) + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln(x^4 + 1) + C
77. ∫ x^2 / (x^4 - 1) dx
a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
b) 1/2 ln|(x^2 + 1)/(x^2 - 1)| + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
78. ∫ x / (x^4 - 1) dx
a) 1/4 ln|(x - 1)/(x + 1)| + C
b) 1/2 ln|(x - 1)/(x + 1)| + C
c) 1/4 ln|(x^2 - 1)| + C
d) 1/2 ln|(x^2 - 1)| + C
Answer: a) 1/4 ln|(x - 1)/(x + 1)| + C
79. ∫ 1 / (x^2 √(x^2 - 1)) dx
a) -√(x^2 - 1) / x + C
b) √(x^2 - 1) / x + C
c) -1 / (x√(x^2 - 1)) + C
d) 1 / (x√(x^2 - 1)) + C
Answer: c) -1 / (x√(x^2 - 1)) + C
80. ∫ x^2 / √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
81. ∫ √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
82. ∫ x / √(x^2 - 1) dx
a) √(x^2 - 1) + C
b) -√(x^2 - 1) + C
c) ln|x + √(x^2 - 1)| + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 - 1) + C
83. ∫ 1 / (x √(x^2 - 1)) dx
a) ln|x + √(x^2 - 1)| + C
b) -ln|x + √(x^2 - 1)| + C
c) cosh^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
84. ∫ x^3 / √(x^2 - 1) dx
a) (x^2 + 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
b) (x^2 - 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
d) (x^2 + 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
Answer: c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
85. ∫ 1 / (x^2 √(1 - x^2)) dx
a) √(1 - x^2) / x + C
b) -√(1 - x^2) / x + C
c) 1 / (x√(1 - x^2)) + C
d) -1 / (x√(1 - x^2)) + C
Answer: d) -1 / (x√(1 - x^2)) + C
86. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
87. ∫ x / √(1 - x^2) dx
a) √(1 - x^2) + C
b) -√(1 - x^2) + C
c) sin^(-1)(x) + C
d) -sin^(-1)(x) + C
Answer: b) -√(1 - x^2) + C
88. ∫ 1 / (x √(1 - x^2)) dx
a) ln|x + √(1 - x^2)| + C
b) -ln|x + √(1 - x^2)| + C
c) sin^(-1)(x) + C
d) cos^(-1)(x) + C
Answer: b) -ln|x + √(1 - x^2)| + C
89. ∫ x^3 / √(1 - x^2) dx
a) -(1 - x^2)^(3/2)/3 + C
b) (1 - x^2)^(3/2)/3 + C
c) -(1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
d) (1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
Answer: a) -(1 - x^2)^(3/2)/3 + C
90. ∫ 1 / (x^2 + x + 1) dx
a) 2/√3 tan^(-1)((2x + 1)/√3) + C
b) 2/√3 tan^(-1)((x + 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x + 1)/√3) + C
d) 1/√3 tan^(-1)((x + 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x + 1)/√3) + C
. ∫ √(x^2 + 1) dx
a) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
Answer: b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
57. ∫ 1 / (x^2 √(x^2 + 1)) dx
a) -√(x^2 + 1) / x + C
b) √(x^2 + 1) / x + C
c) -1 / (x√(x^2 + 1)) + C
d) 1 / (x√(x^2 + 1)) + C
Answer: c) -1 / (x√(x^2 + 1)) + C
58. ∫ x / (x^2 + 4)^2 dx
a) -1 / (4(x^2 + 4)) + C
b) 1 / (4(x^2 + 4)) + C
c) -x / (2(x^2 + 4)) + C
d) x / (2(x^2 + 4)) + C
Answer: a) -1 / (4(x^2 + 4)) + C
59. ∫ x^2 / (x^2 + 4)^2 dx
a) x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
c) -x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
d) -x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
Answer: b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
60. ∫ x^3 / (x^2 + 4)^2 dx
a) (x^2 - 4) / (4(x^2 + 4)) + C
b) (x^2 + 4) / (4(x^2 + 4)) + C
c) (x^2 - 4) / (2(x^2 + 4)) + C
d) (x^2 + 4) / (2(x^2 + 4)) + C
Answer: a) (x^2 - 4) / (4(x^2 + 4)) + C
61. ∫ sin(ln(x)) / x dx
a) -cos(ln(x))/2 + sin(ln(x))/2 + C
b) cos(ln(x))/2 - sin(ln(x))/2 + C
c) -cos(ln(x))/2 - sin(ln(x))/2 + C
d) cos(ln(x))/2 + sin(ln(x))/2 + C
Answer: a) -cos(ln(x))/2 + sin(ln(x))/2 + C
62. ∫ cos(ln(x)) / x dx
a) sin(ln(x))/2 + cos(ln(x))/2 + C
b) sin(ln(x))/2 - cos(ln(x))/2 + C
c) -sin(ln(x))/2 + cos(ln(x))/2 + C
d) -sin(ln(x))/2 - cos(ln(x))/2 + C
Answer: a) sin(ln(x))/2 + cos(ln(x))/2 + C
63. ∫ tan(ln(x)) / x dx
a) -ln|cos(ln(x))| + C
b) ln|cos(ln(x))| + C
c) ln|sin(ln(x))| + C
d) -ln|sin(ln(x))| + C
Answer: b) ln|cos(ln(x))| + C
64. ∫ sec(ln(x)) / x dx
a) tan(ln(x)) + C
b) -tan(ln(x)) + C
c) cot(ln(x)) + C
d) -cot(ln(x)) + C
Answer: a) tan(ln(x)) + C
65. ∫ csc(ln(x)) / x dx
a) ln|csc(ln(x)) - cot(ln(x))| + C
b) -ln|csc(ln(x)) - cot(ln(x))| + C
c) ln|tan(ln(x)/2)| + C
d) -ln|tan(ln(x)/2)| + C
Answer: d) -ln|tan(ln(x)/2)| + C
66. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) -cos(x^3)/3 + C
c) sin(x^3)/3 + C
d) cos(x^3)/3 + C
Answer: b) -cos(x^3)/3 + C
67. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) -1/2 sin(x^2) + C
c) cos(x^2)/2 + C
d) -cos(x^2)/2 + C
Answer: a) 1/2 sin(x^2) + C
68. ∫ e^(sin(x)) cos(x) dx
a) e^(sin(x)) + C
b) -e^(sin(x)) + C
c) e^(-sin(x)) + C
d) -e^(-sin(x)) + C
Answer: a) e^(sin(x)) + C
69. ∫ x / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: d) 1/4 tan^(-1)(x^2) + C
70. ∫ x^3 / (x^4 + 1) dx
a) 1/2 ln(x^4 + 1) + C
b) 1/4 ln(x^4 + 1) + C
c) 1/2 tan^(-1)(x^2) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: b) 1/4 ln(x^4 + 1) + C
71. ∫ x^2 / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
72. ∫ 1 / (x^3 + x) dx
a) 1/2 ln|x^2 - x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
c) 1/2 ln|x^2 - x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/2 ln|x^2 + x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
73. ∫ x^2 / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) ln|x + 1| - (1/√3) tan^(-1)((2x - 1)/√3) + C
c) ln|x + 1| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|x^3 - 1| + C
Answer: a) 1/3 ln|x^3 + 1| + C
74. ∫ x / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
c) 1/3 ln|(x + 1)/(x^2 - x + 1)| - (2/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|(x - 1)/(x^2 + x + 1)| + (2/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
75. ∫ 1 / (x^2 (x^2 + 1)) dx
a) -1/x + tan^(-1)(x) + C
b) 1/x - tan^(-1)(x) + C
c) -1/x - tan^(-1)(x) + C
d) 1/x + tan^(-1)(x) + C
Answer: c) -1/x - tan^(-1)(x) + C
76. ∫ x^3 / (x^4 + 1) dx
a) 1/4 ln(x^4 + 1) + C
b) 1/2 ln(x^4 + 1) + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln(x^4 + 1) + C
77. ∫ x^2 / (x^4 - 1) dx
a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
b) 1/2 ln|(x^2 + 1)/(x^2 - 1)| + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
78. ∫ x / (x^4 - 1) dx
a) 1/4 ln|(x - 1)/(x + 1)| + C
b) 1/2 ln|(x - 1)/(x + 1)| + C
c) 1/4 ln|(x^2 - 1)| + C
d) 1/2 ln|(x^2 - 1)| + C
Answer: a) 1/4 ln|(x - 1)/(x + 1)| + C
79. ∫ 1 / (x^2 √(x^2 - 1)) dx
a) -√(x^2 - 1) / x + C
b) √(x^2 - 1) / x + C
c) -1 / (x√(x^2 - 1)) + C
d) 1 / (x√(x^2 - 1)) + C
Answer: c) -1 / (x√(x^2 - 1)) + C
80. ∫ x^2 / √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
81. ∫ √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
82. ∫ x / √(x^2 - 1) dx
a) √(x^2 - 1) + C
b) -√(x^2 - 1) + C
c) ln|x + √(x^2 - 1)| + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 - 1) + C
83. ∫ 1 / (x √(x^2 - 1)) dx
a) ln|x + √(x^2 - 1)| + C
b) -ln|x + √(x^2 - 1)| + C
c) cosh^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
84. ∫ x^3 / √(x^2 - 1) dx
a) (x^2 + 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
b) (x^2 - 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
d) (x^2 + 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
Answer: c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
85. ∫ 1 / (x^2 √(1 - x^2)) dx
a) √(1 - x^2) / x + C
b) -√(1 - x^2) / x + C
c) 1 / (x√(1 - x^2)) + C
d) -1 / (x√(1 - x^2)) + C
Answer: d) -1 / (x√(1 - x^2)) + C
86. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
87. ∫ x / √(1 - x^2) dx
a) √(1 - x^2) + C
b) -√(1 - x^2) + C
c) sin^(-1)(x) + C
d) -sin^(-1)(x) + C
Answer: b) -√(1 - x^2) + C
88. ∫ 1 / (x √(1 - x^2)) dx
a) ln|x + √(1 - x^2)| + C
b) -ln|x + √(1 - x^2)| + C
c) sin^(-1)(x) + C
d) cos^(-1)(x) + C
Answer: b) -ln|x + √(1 - x^2)| + C
89. ∫ x^3 / √(1 - x^2) dx
a) -(1 - x^2)^(3/2)/3 + C
b) (1 - x^2)^(3/2)/3 + C
c) -(1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
d) (1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
Answer: a) -(1 - x^2)^(3/2)/3 + C
90. ∫ 1 / (x^2 + x + 1) dx
a) 2/√3 tan^(-1)((2x + 1)/√3) + C
b) 2/√3 tan^(-1)((x + 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x + 1)/√3) + C
d) 1/√3 tan^(-1)((x + 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x + 1)/√3) + C
. ∫ √(x^2 + 1) dx
a) x√(x^2 + 1)/2 + ln(x + √(x^2 + 1))/2 + C
b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
c) x√(x^2 + 1)/2 - sinh^(-1)(x)/2 + C
d) x√(x^2 + 1)/2 - ln(x + √(x^2 + 1))/2 + C
Answer: b) x√(x^2 + 1)/2 + sinh^(-1)(x)/2 + C
57. ∫ 1 / (x^2 √(x^2 + 1)) dx
a) -√(x^2 + 1) / x + C
b) √(x^2 + 1) / x + C
c) -1 / (x√(x^2 + 1)) + C
d) 1 / (x√(x^2 + 1)) + C
Answer: c) -1 / (x√(x^2 + 1)) + C
58. ∫ x / (x^2 + 4)^2 dx
a) -1 / (4(x^2 + 4)) + C
b) 1 / (4(x^2 + 4)) + C
c) -x / (2(x^2 + 4)) + C
d) x / (2(x^2 + 4)) + C
Answer: a) -1 / (4(x^2 + 4)) + C
59. ∫ x^2 / (x^2 + 4)^2 dx
a) x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
c) -x / (4(x^2 + 4)) + 1/8 tan^(-1)(x/2) + C
d) -x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
Answer: b) x / (4(x^2 + 4)) - 1/8 tan^(-1)(x/2) + C
60. ∫ x^3 / (x^2 + 4)^2 dx
a) (x^2 - 4) / (4(x^2 + 4)) + C
b) (x^2 + 4) / (4(x^2 + 4)) + C
c) (x^2 - 4) / (2(x^2 + 4)) + C
d) (x^2 + 4) / (2(x^2 + 4)) + C
Answer: a) (x^2 - 4) / (4(x^2 + 4)) + C
61. ∫ sin(ln(x)) / x dx
a) -cos(ln(x))/2 + sin(ln(x))/2 + C
b) cos(ln(x))/2 - sin(ln(x))/2 + C
c) -cos(ln(x))/2 - sin(ln(x))/2 + C
d) cos(ln(x))/2 + sin(ln(x))/2 + C
Answer: a) -cos(ln(x))/2 + sin(ln(x))/2 + C
62. ∫ cos(ln(x)) / x dx
a) sin(ln(x))/2 + cos(ln(x))/2 + C
b) sin(ln(x))/2 - cos(ln(x))/2 + C
c) -sin(ln(x))/2 + cos(ln(x))/2 + C
d) -sin(ln(x))/2 - cos(ln(x))/2 + C
Answer: a) sin(ln(x))/2 + cos(ln(x))/2 + C
63. ∫ tan(ln(x)) / x dx
a) -ln|cos(ln(x))| + C
b) ln|cos(ln(x))| + C
c) ln|sin(ln(x))| + C
d) -ln|sin(ln(x))| + C
Answer: b) ln|cos(ln(x))| + C
64. ∫ sec(ln(x)) / x dx
a) tan(ln(x)) + C
b) -tan(ln(x)) + C
c) cot(ln(x)) + C
d) -cot(ln(x)) + C
Answer: a) tan(ln(x)) + C
65. ∫ csc(ln(x)) / x dx
a) ln|csc(ln(x)) - cot(ln(x))| + C
b) -ln|csc(ln(x)) - cot(ln(x))| + C
c) ln|tan(ln(x)/2)| + C
d) -ln|tan(ln(x)/2)| + C
Answer: d) -ln|tan(ln(x)/2)| + C
66. ∫ x^2 sin(x^3) dx
a) -1/3 cos(x^3) + C
b) -cos(x^3)/3 + C
c) sin(x^3)/3 + C
d) cos(x^3)/3 + C
Answer: b) -cos(x^3)/3 + C
67. ∫ x cos(x^2) dx
a) 1/2 sin(x^2) + C
b) -1/2 sin(x^2) + C
c) cos(x^2)/2 + C
d) -cos(x^2)/2 + C
Answer: a) 1/2 sin(x^2) + C
68. ∫ e^(sin(x)) cos(x) dx
a) e^(sin(x)) + C
b) -e^(sin(x)) + C
c) e^(-sin(x)) + C
d) -e^(-sin(x)) + C
Answer: a) e^(sin(x)) + C
69. ∫ x / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: d) 1/4 tan^(-1)(x^2) + C
70. ∫ x^3 / (x^4 + 1) dx
a) 1/2 ln(x^4 + 1) + C
b) 1/4 ln(x^4 + 1) + C
c) 1/2 tan^(-1)(x^2) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: b) 1/4 ln(x^4 + 1) + C
71. ∫ x^2 / (x^4 + 1) dx
a) 1/2 tan^(-1)(x^2) + C
b) 1/4 ln((x^2 + 1)/(x^2 - 1)) + C
c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
d) 1/4 tan^(-1)(x^2) + C
Answer: c) 1/2 ln((x^2 + 1)/(x^2 - 1)) + C
72. ∫ 1 / (x^3 + x) dx
a) 1/2 ln|x^2 - x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
c) 1/2 ln|x^2 - x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/2 ln|x^2 + x + 1| + 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/2 ln|x^2 + x + 1| - 1/2 ln|x| + (1/√3) tan^(-1)((2x + 1)/√3) + C
73. ∫ x^2 / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) ln|x + 1| - (1/√3) tan^(-1)((2x - 1)/√3) + C
c) ln|x + 1| + (1/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|x^3 - 1| + C
Answer: a) 1/3 ln|x^3 + 1| + C
74. ∫ x / (x^3 + 1) dx
a) 1/3 ln|x^3 + 1| + C
b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
c) 1/3 ln|(x + 1)/(x^2 - x + 1)| - (2/√3) tan^(-1)((2x - 1)/√3) + C
d) 1/3 ln|(x - 1)/(x^2 + x + 1)| + (2/√3) tan^(-1)((2x + 1)/√3) + C
Answer: b) 1/3 ln|(x + 1)/(x^2 - x + 1)| + (2/√3) tan^(-1)((2x - 1)/√3) + C
75. ∫ 1 / (x^2 (x^2 + 1)) dx
a) -1/x + tan^(-1)(x) + C
b) 1/x - tan^(-1)(x) + C
c) -1/x - tan^(-1)(x) + C
d) 1/x + tan^(-1)(x) + C
Answer: c) -1/x - tan^(-1)(x) + C
76. ∫ x^3 / (x^4 + 1) dx
a) 1/4 ln(x^4 + 1) + C
b) 1/2 ln(x^4 + 1) + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln(x^4 + 1) + C
77. ∫ x^2 / (x^4 - 1) dx
a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
b) 1/2 ln|(x^2 + 1)/(x^2 - 1)| + C
c) 1/4 tan^(-1)(x^2) + C
d) 1/2 tan^(-1)(x^2) + C
Answer: a) 1/4 ln|(x^2 + 1)/(x^2 - 1)| + C
78. ∫ x / (x^4 - 1) dx
a) 1/4 ln|(x - 1)/(x + 1)| + C
b) 1/2 ln|(x - 1)/(x + 1)| + C
c) 1/4 ln|(x^2 - 1)| + C
d) 1/2 ln|(x^2 - 1)| + C
Answer: a) 1/4 ln|(x - 1)/(x + 1)| + C
79. ∫ 1 / (x^2 √(x^2 - 1)) dx
a) -√(x^2 - 1) / x + C
b) √(x^2 - 1) / x + C
c) -1 / (x√(x^2 - 1)) + C
d) 1 / (x√(x^2 - 1)) + C
Answer: c) -1 / (x√(x^2 - 1)) + C
80. ∫ x^2 / √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
81. ∫ √(x^2 - 1) dx
a) x√(x^2 - 1)/2 + ln|x + √(x^2 - 1)|/2 + C
b) x√(x^2 - 1)/2 - ln|x + √(x^2 - 1)|/2 + C
c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
d) x√(x^2 - 1)/2 - cosh^(-1)(x)/2 + C
Answer: c) x√(x^2 - 1)/2 + cosh^(-1)(x)/2 + C
82. ∫ x / √(x^2 - 1) dx
a) √(x^2 - 1) + C
b) -√(x^2 - 1) + C
c) ln|x + √(x^2 - 1)| + C
d) cosh^(-1)(x) + C
Answer: a) √(x^2 - 1) + C
83. ∫ 1 / (x √(x^2 - 1)) dx
a) ln|x + √(x^2 - 1)| + C
b) -ln|x + √(x^2 - 1)| + C
c) cosh^(-1)(x) + C
d) sec^(-1)(x) + C
Answer: d) sec^(-1)(x) + C
84. ∫ x^3 / √(x^2 - 1) dx
a) (x^2 + 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
b) (x^2 - 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
d) (x^2 + 1)√(x^2 - 1)/3 + 2/3 cosh^(-1)(x) + C
Answer: c) (x^2 - 1)√(x^2 - 1)/3 - 2/3 cosh^(-1)(x) + C
85. ∫ 1 / (x^2 √(1 - x^2)) dx
a) √(1 - x^2) / x + C
b) -√(1 - x^2) / x + C
c) 1 / (x√(1 - x^2)) + C
d) -1 / (x√(1 - x^2)) + C
Answer: d) -1 / (x√(1 - x^2)) + C
86. ∫ x^2 / √(1 - x^2) dx
a) x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
b) x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
d) -x√(1 - x^2)/2 - sin^(-1)(x)/2 + C
Answer: c) -x√(1 - x^2)/2 + sin^(-1)(x)/2 + C
87. ∫ x / √(1 - x^2) dx
a) √(1 - x^2) + C
b) -√(1 - x^2) + C
c) sin^(-1)(x) + C
d) -sin^(-1)(x) + C
Answer: b) -√(1 - x^2) + C
88. ∫ 1 / (x √(1 - x^2)) dx
a) ln|x + √(1 - x^2)| + C
b) -ln|x + √(1 - x^2)| + C
c) sin^(-1)(x) + C
d) cos^(-1)(x) + C
Answer: b) -ln|x + √(1 - x^2)| + C
89. ∫ x^3 / √(1 - x^2) dx
a) -(1 - x^2)^(3/2)/3 + C
b) (1 - x^2)^(3/2)/3 + C
c) -(1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
d) (1 - x^2)^(3/2)/3 - x√(1 - x^2) + C
Answer: a) -(1 - x^2)^(3/2)/3 + C
90. ∫ 1 / (x^2 + x + 1) dx
a) 2/√3 tan^(-1)((2x + 1)/√3) + C
b) 2/√3 tan^(-1)((x + 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x + 1)/√3) + C
d) 1/√3 tan^(-1)((x + 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x + 1)/√3) + C
91. ∫ x / (x^2 + x + 1) dx
a) 1/2 ln(x^2 + x + 1) - 1/√3 tan^(-1)((2x + 1)/√3) + C
b) 1/2 ln(x^2 + x + 1) + 1/√3 tan^(-1)((2x + 1)/√3) + C
c) ln(x^2 + x + 1) - 2/√3 tan^(-1)((2x + 1)/√3) + C
d) ln(x^2 + x + 1) + 2/√3 tan^(-1)((2x + 1)/√3) + C
Answer: a) 1/2 ln(x^2 + x + 1) - 1/√3 tan^(-1)((2x + 1)/√3) + C
92. ∫ x^2 / (x^2 + x + 1) dx
a) x - 1/2 ln(x^2 + x + 1) + √3/2 tan^(-1)((2x + 1)/√3) + C
b) x + 1/2 ln(x^2 + x + 1) - √3/2 tan^(-1)((2x + 1)/√3) + C
c) x - 1/2 ln(x^2 + x + 1) - √3/2 tan^(-1)((2x + 1)/√3) + C
d) x + 1/2 ln(x^2 + x + 1) + √3/2 tan^(-1)((2x + 1)/√3) + C
Answer: b) x + 1/2 ln(x^2 + x + 1) - √3/2 tan^(-1)((2x + 1)/√3) + C
93. ∫ 1 / (x^2 - x + 1) dx
a) 2/√3 tan^(-1)((2x - 1)/√3) + C
b) 2/√3 tan^(-1)((x - 1/2)/√3/2) + C
c) 1/√3 tan^(-1)((2x - 1)/√3) + C
d) 1/√3 tan^(-1)((x - 1/2)/√3/2) + C
Answer: a) 2/√3 tan^(-1)((2x - 1)/√3) + C
94. ∫ x / (x^2 - x + 1) dx
a) 1/2 ln(x^2 - x + 1) - 1/√3 tan^(-1)((2x - 1)/√3) + C
b) 1/2 ln(x^2 - x + 1) + 1/√3 tan^(-1)((2x - 1)/√3) + C
c) ln(x^2 - x + 1) - 2/√3 tan^(-1)((2x - 1)/√3) + C
d) ln(x^2 - x + 1) + 2/√3 tan^(-1)((2x - 1)/√3) + C
Answer: b) 1/2 ln(x^2 - x + 1) + 1/√3 tan^(-1)((2x - 1)/√3) + C
95. ∫ x^2 / (x^2 - x + 1) dx
a) x - 1/2 ln(x^2 - x + 1) + √3/2 tan^(-1)((2x - 1)/√3) + C
b) x + 1/2 ln(x^2 - x + 1) - √3/2 tan^(-1)((2x - 1)/√3) + C
c) x - 1/2 ln(x^2 - x + 1) - √3/2 tan^(-1)((2x - 1)/√3) + C
d) x + 1/2 ln(x^2 -
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