ADVANCED EXAM INTEGRATION
QUESTION BANK WITH ANSWERS- SET1
1. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥,
𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥
= ∫ sin𝑥
1 + 𝑢2 𝑑𝑢
= arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
2. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢
= 𝑥3+ 1,
𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 =
1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
3. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢
=ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 =
∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
4. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥,
𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥
= ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
5. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥
=ln|sin𝑥|+ 𝐶
6. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥
= − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
7. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥
=ln|1 + 𝑒𝑥|+ 𝐶
8. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
9. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
10. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function
11. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
12. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
13. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
14. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
15. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
16. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
17. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
18. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
19. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
20. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
21. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
22. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
23. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
24. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
25. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
26. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
27. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
28. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
29. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
30. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
31. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
32. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
33. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
34. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
35. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
36. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
37. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
38. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
39. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
40. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
41. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
42. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
43. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
44. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
45. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
46. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
47. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
48. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
49. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
50. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
51. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
52. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
53. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
54. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
55. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
56. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
57. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
58. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
59. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
60. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
61. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
62. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
63. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
64. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
65. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
66. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
67. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
68. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
69. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
70. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
71. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
72. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
73. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
74. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
75. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
76. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
77. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
78. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
79. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
80. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
81. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
82. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
83. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
84. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
85. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
86. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
87. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
88. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
89. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
90. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
91. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
92. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
93. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
94. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
95. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
96. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
97. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
98. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
99. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
100. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
101. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
102. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
103. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
104. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
105. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
106. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
107. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
108. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
109. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
110. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
111. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
112. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
113. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
114. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
115. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
116. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
117. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
118. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
119. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
120. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
121. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
122. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
123. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
124. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
125. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
126. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
127. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
128. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
129. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
130. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
131. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
132. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
133. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
134. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
135. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
136. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
137. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
138. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
139. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
140. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
141. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
142. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
143. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
144. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
145. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
146. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
147. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
148. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
149. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
150. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
151. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
152. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
153. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
154. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
155. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
156. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
157. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
158. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
159. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
160. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
161. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
162. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
163. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
164. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
165. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
166. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
167. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
168. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
169. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
170. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
171. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
172. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
173. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
174. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
175. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
176. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
177. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
178. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
179. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
180. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
181. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
182. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
183. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
184. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
185. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
186. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
187. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
188. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
189. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
190. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
191. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
192. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
193. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
194. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
195. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
196. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
197. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
198. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
199. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
200. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
201. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
202. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
203. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
204. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
205. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
206. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
207. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
208. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
209. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
210. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
211. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
212. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
213. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
214. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
215. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
216. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
217. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
218. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
219. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
220. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
221. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
222. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
223. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
224. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
225. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
226. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
227. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
228. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
229. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
230. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
231. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
232. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
233. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
234. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
235. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
236. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
237. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
238. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
239. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
240. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
241. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
242. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
243. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
244. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
245. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
246. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
247. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
248. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
249. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
250. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
251. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
252. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
253. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
254. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
255. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
256. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
257. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
258. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
259. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
260. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
261. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
262. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
263. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
264. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
265. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
266. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
267. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
268. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
269. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
270. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
271. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
272. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
273. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
274. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
275. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
276. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
277. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
278. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
279. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
280. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
281. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
282. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
283. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
284. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
285. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
286. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
287. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
288. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
289. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
290. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
291. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
292. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
293. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
294. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
295. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
296. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
297. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
298. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
299. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
300. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
301. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
302. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
303. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
304. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
305. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
306. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
307. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
308. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
309. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
310. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
311. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
312. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
313. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
314. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
315. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
316. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
317. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
318. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
319. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
320. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
321. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
322. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
323. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
324. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
325. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
326. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
327. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
328. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
329. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
330. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
331. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
332. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
333. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
334. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
335. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
336. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
337. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
338. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
339. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
340. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
341. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
342. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
343. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
344. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
345. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
346. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
347. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
348. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
349. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
350. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
351. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
352. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
353. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
354. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
355. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
356. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
357. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
358. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
359. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
360. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
361. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
362. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
363. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
364. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
365. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
366. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
367. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
368. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
369. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
370. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
371. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
372. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
373. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
374. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
375. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
376. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
377. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
378. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
379. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
380. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
381. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
382. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
383. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
384. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
385. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
386. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
387. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
388. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
389. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
390. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
391. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
392. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
393. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
394. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
395. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
396. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
397. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
398. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
399. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
400. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
401. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
402. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
403. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
404. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
405. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
406. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
407. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
408. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
409. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
410. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
411. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
412. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
413. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
414. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
415. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
416. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
417. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
418. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
419. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
420. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
421. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
422. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
423. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
424. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
425. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
426. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
427. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
428. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
429. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
430. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
431. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
432. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
433. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
434. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
435. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
436. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
437. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
438. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
439. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
440. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
441. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
442. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
443. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
444. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
445. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
446. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
447. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
448. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
449. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
450. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
451. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
452. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
453. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
454. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
455. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
456. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
457. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
458. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
459. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
460. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
461. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
462. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
463. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
464. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
465. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
466. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
467. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
468. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
469. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
470. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
471. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
472. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
473. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
474. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
475. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
476. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
477. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
478. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
479. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
480. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
481. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
482. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
483. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
484. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
485. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
486. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
487. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
488. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
489. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
490. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
491. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
492. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
493. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
494. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
495. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
496. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
497. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
498. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
499. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
500. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
501. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
502. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
503. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
504. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
505. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
506. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
507. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
508. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
509. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
510. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
511. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
512. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
513. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
514. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
515. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
516. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
517. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
518. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
519. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
520. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
521. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
522. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
523. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
524. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
525. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
526. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
527. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
528. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
529. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
530. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
531. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
532. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
533. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
534. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
535. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
536. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
537. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
538. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
539. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
540. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
541. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
542. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
543. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
544. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
545. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
546. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
547. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
548. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
549. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
550. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
551. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
552. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
553. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
554. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
555. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
556. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
557. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
558. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
559. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
560. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
561. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
562. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
563. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
564. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
565. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
566. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
567. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
568. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
569. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
570. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
571. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
572. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
573. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
574. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
575. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
576. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
577. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
578. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
579. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
580. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
581. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
582. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
583. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
584. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
585. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
586. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
587. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
588. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
589. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
590. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
591. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
592. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
593. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
594. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
595. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
596. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
597. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
598. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
599. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
600. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
601. Evaluate ∫𝑒𝑥sin𝑥
1+𝑒2𝑥 𝑑𝑥.
Solution:
Let 𝑢 = 𝑒𝑥, 𝑑𝑢 = 𝑒𝑥 𝑑𝑥.
∫𝑒𝑥sin𝑥
1 + 𝑒2𝑥 𝑑𝑥 = ∫ sin𝑥
1 + 𝑢2 𝑑𝑢 = arctan(𝑢)+ 𝐶 = arctan(𝑒𝑥)+ 𝐶
602. Evaluate ∫𝑥2
(𝑥3+1)2 𝑑𝑥.
Solution:
Let 𝑢 = 𝑥3+ 1, 𝑑𝑢 = 3𝑥2 𝑑𝑥.
∫𝑥2
(𝑥3+ 1)2 𝑑𝑥 = 1
3∫1
𝑢2 𝑑𝑢 = − 1
3(𝑥3+ 1)+ 𝐶
603. Evaluate ∫ln𝑥
𝑥 𝑑𝑥.
Solution:
Let 𝑢 = ln𝑥, 𝑑𝑢 = 1
𝑥 𝑑𝑥.
∫ln𝑥
𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (ln𝑥)2
2+ 𝐶
604. Evaluate ∫sin−1𝑥
√1−𝑥2 𝑑𝑥.
Solution:
Let 𝑢 = sin−1𝑥, 𝑑𝑢 = 1
√1 − 𝑥2 𝑑𝑥.
∫sin−1𝑥
√1 − 𝑥2 𝑑𝑥 = ∫ 𝑢 𝑑𝑢 = 𝑢2
2+ 𝐶 = (sin−1𝑥)2
2+ 𝐶
605. Evaluate ∫𝑥cos𝑥
sin2𝑥 𝑑𝑥.
Solution:
∫𝑥cos𝑥
sin2𝑥 𝑑𝑥 = ln|sin𝑥|+ 𝐶
606. Evaluate ∫√1−𝑥2
𝑥2 𝑑𝑥.
Solution:
∫√1 − 𝑥2
𝑥2 𝑑𝑥 = − √1 − 𝑥2
𝑥− arcsin𝑥 + 𝐶
607. Evaluate ∫𝑒𝑥
1+𝑒𝑥 𝑑𝑥.
Solution:
∫𝑒𝑥
1 + 𝑒𝑥 𝑑𝑥 = ln|1 + 𝑒𝑥|+ 𝐶
608. Evaluate ∫cos𝑥
√1−sin𝑥 𝑑𝑥.
Solution:
∫cos𝑥
√1 − sin𝑥 𝑑𝑥 = 2√1 − sin𝑥 + 𝐶
609. Evaluate ∫𝑥2
√1+𝑥3 𝑑𝑥.
Solution:
∫𝑥2
√1 + 𝑥3 𝑑𝑥 = 2
3(1 + 𝑥3)3/2 + 𝐶
610. Evaluate ∫ln(1+𝑥)
𝑥 𝑑𝑥.
Solution:
∫ln(1 + 𝑥)
𝑥 𝑑𝑥 = Li2(−𝑥)+(ln(1 + 𝑥))2
2+ 𝐶
where Li2(−𝑥) is the polylogarithm function.
