Indefinite Integrals: 100 items Problem Set Homework – Solutions

1. ∫ 4 dx = 4x + C

2. ∫ x^6 dx = x^7/7 + C

3. ∫ (5x^4 – 3x^2 + 1) dx = x^5 – x^3 + x + C

4. ∫ (2/x + e^x) dx = 2ln|x| + e^x + C

5. ∫ x^(3/2) dx = (2/5)x^(5/2) + C

6. ∫ 10x^9 dx = x^10 + C

7. ∫ (x^3 + 6x^2 – 4x + 5) dx = x^4/4 + 2x^3 – 2x^2 + 5x + C

8. ∫ (1/x^4) dx = ∫ x^-4 dx = -1/(3x^3) + C

9. ∫ e^(2x) dx = (1/2)e^(2x) + C

10. ∫ sin(3x) dx = -(1/3)cos(3x) + C

11. ∫ 7x^2 dx = (7/3)x^3 + C

12. ∫ (4x^3 + 2x – 1) dx = x^4 + x^2 – x + C

13. ∫ (3/x^2) dx = ∫ 3x^-2 dx = -3/x + C

14. ∫ 2e^(5x) dx = (2/5)e^(5x) + C

15. ∫ cos(4x) dx = (1/4)sin(4x) + C

16. ∫ x^(1/3) dx = (3/4)x^(4/3) + C

17. ∫ (6x^5 – 4x^3 + 8x) dx = x^6 – x^4 + 4x^2 + C

18. ∫ (5/x) dx = 5ln|x| + C

19. ∫ -3e^(-x) dx = 3e^(-x) + C

20. ∫ -2sin(2x) dx = cos(2x) + C

21. ∫ (x^4 + 3x^2 + 1) dx = x^5/5 + x^3 + x + C

22. ∫ (1/(2x)) dx = (1/2)ln|x| + C

23. ∫ 4e^(x/2) dx = 8e^(x/2) + C

24. ∫ 3cos(x/3) dx = 9sin(x/3) + C

25. ∫ x^(-2/3) dx = 3x^(1/3) + C

26. ∫ (8x^7 – 6x^5 + 10x^3 – 2) dx = x^8 – x^6 + (5/2)x^4 – 2x + C

27. ∫ (7/x^3) dx = ∫ 7x^-3 dx = -7/(2x^2) + C

28. ∫ 5e^(2x + 1) dx = (5/2)e^(2x + 1) + C

29. ∫ (sin(x/2) + cos(2x)) dx = -2cos(x/2) + (1/2)sin(2x) + C

30. ∫ (x^2 + 1)^2 dx = ∫ (x^4 + 2x^2 + 1) dx = x^5/5 + (2/3)x^3 + x + C

31. ∫ x(x^2 + 1)^3 dx = Let u = x^2 + 1, du = 2x dx => (1/2) ∫ u^3 du = (1/8)u^4 + C = (1/8)(x^2 + 1)^4 + C

32. ∫ (x + 1) / x dx = ∫ (1 + 1/x) dx = x + ln|x| + C

33. ∫ (x^2 – 4) / (x – 2) dx = ∫ (x + 2) dx = x^2/2 + 2x + C

34. ∫ x / (x + 1) dx = ∫ (1 – 1/(x + 1)) dx = x – ln|x + 1| + C

35. ∫ 1 / (x ln(x)) dx = Let u = ln(x), du = (1/x) dx => ∫ 1/u du = ln|u| + C = ln|ln(x)| + C

36. ∫ x e^(x^2) dx = Let u = x^2, du = 2x dx => (1/2) ∫ e^u du = (1/2)e^u + C = (1/2)e^(x^2) + C

37. ∫ sin^2(x) cos(x) dx = Let u = sin(x), du = cos(x) dx => ∫ u^2 du = u^3/3 + C = (1/3)sin^3(x) + C

38. ∫ cos^3(x) sin(x) dx = Let u = cos(x), du = -sin(x) dx => -∫ u^3 du = -(1/4)u^4 + C = -(1/4)cos^4(x) + C

39. ∫ tan(x) sec^2(x) dx = Let u = tan(x), du = sec^2(x) dx => ∫ u du = u^2/2 + C = (1/2)tan^2(x) + C

40. ∫ cot(x) csc^2(x) dx = Let u = cot(x), du = -csc^2(x) dx => -∫ u du = -(1/2)u^2 + C = -(1/2)cot^2(x) + C

41. ∫ (x^2 + 1) / (x^3 + 3x) dx = Let u = x^3 + 3x, du = (3x^2 + 3) dx => (1/3) ∫ 1/u du = (1/3)ln|u| + C = (1/3)ln|x^3 + 3x| + C

42. ∫ 1 / (x^2 + 4) dx = (1/2)arctan(x/2) + C

43. ∫ 1 / √(1 – x^2) dx = arcsin(x) + C

44. ∫ 1 / (√(x + 1) + 1) dx = Let u = √(x + 1), x = u^2 – 1, dx = 2u du => ∫ 2u/(u + 1) du = 2∫ (1 – 1/(u + 1)) du = 2(u – ln|u + 1|) + C = 2(√(x + 1) – ln(√(x + 1) + 1)) + C

45. ∫ x / √(2x + 1) dx = Let u = √(2x + 1), x = (u^2 – 1)/2, dx = u du => ∫ (u^2 – 1)/(2u) * u du = (1/2) ∫ (u^2 – 1) du = (1/6)u^3 – (1/2)u + C = (1/6)(2x + 1)^(3/2) – (1/2)√(2x + 1) + C

46. ∫ (e^x + 1)^2 e^x dx = Let u = e^x + 1, du = e^x dx => ∫ u^2 du = u^3/3 + C = (1/3)(e^x + 1)^3 + C

47. ∫ e^(e^x) e^x dx = Let u = e^x, du = e^x dx => ∫ e^u du = e^u + C = e^(e^x) + C

48. ∫ (ln x) / x dx = Let u = ln x, du = (1/x) dx => ∫ u du = u^2/2 + C = (1/2)(ln x)^2 + C

49. ∫ x^3 / (x^2 + 1)^(3/2) dx = Let u = x^2 + 1, x^2 = u – 1, 2x dx = du => ∫ (u – 1) / u^(3/2) * (1/2) du = (1/2) ∫ (u^(-1/2) – u^(-3/2)) du = u^(1/2) + u^(-1/2) + C = √(x^2 + 1) + 1/√(x^2 + 1) + C

50. ∫ (arcsin x) / √(1 – x^2) dx = Let u = arcsin x, du = 1/√(1 – x^2) dx => ∫ u du = u^2/2 + C = (1/2)(arcsin x)^2 + C

Here are the step-by-step solutions for the indefinite integral problems 51 – 100:

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