Table of Integrals

Basic Integrals

1. \int {u}^{n}\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1

2. \int \frac{du}{u}=\text{ln}\phantom{\rule{0.1em}{0ex}}|u|+C

3. \int {e}^{u}\phantom{\rule{0.2em}{0ex}}du={e}^{u}+C

4. \int {a}^{u}\phantom{\rule{0.2em}{0ex}}du=\frac{{a}^{u}}{\text{ln}\phantom{\rule{0.1em}{0ex}}a}+C

5. \int \text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−cos}\phantom{\rule{0.2em}{0ex}}u+C

6. \int \text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sin}\phantom{\rule{0.2em}{0ex}}u+C

7. \int {\text{sec}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tan}\phantom{\rule{0.2em}{0ex}}u+C

8. \int {\text{csc}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−cot}\phantom{\rule{0.2em}{0ex}}u+C

9. \int \text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sec}\phantom{\rule{0.2em}{0ex}}u+C

10. \int \text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−csc}\phantom{\rule{0.2em}{0ex}}u+C

11. \int \text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u|+C

12. \int \text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sin}\phantom{\rule{0.2em}{0ex}}u|+C

13. \int \text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u+\text{tan}\phantom{\rule{0.2em}{0ex}}u|+C

14. \int \text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{csc}\phantom{\rule{0.2em}{0ex}}u-\text{cot}\phantom{\rule{0.2em}{0ex}}u|+C

15. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}-{u}^{2}}}={\text{sin}}^{-1}\frac{u}{a}+C

16. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{a}^{2}+{u}^{2}}=\frac{1}{a}{\text{tan}}^{-1}\frac{u}{a}+C

17. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{u}^{2}-{a}^{2}}}=\frac{1}{a}{\text{sec}}^{-1}\frac{u}{a}+C

Trigonometric Integrals

18. \int {\text{sin}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}u-\frac{1}{4}\text{sin}\phantom{\rule{0.2em}{0ex}}2u+C

19. \int {\text{cos}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}u+\frac{1}{4}\text{sin}\phantom{\rule{0.2em}{0ex}}2u+C

20. \int {\text{tan}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tan}\phantom{\rule{0.2em}{0ex}}u-u+C

21. \int {\text{cot}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{cot}\phantom{\rule{0.2em}{0ex}}u-u+C

22. \int {\text{sin}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{3}\left(2+{\text{sin}}^{2}u\right)\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+C

23. \int {\text{cos}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{3}\left(2+{\text{cos}}^{2}u\right)\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+C

24. \int {\text{tan}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}{\text{tan}}^{2}u+\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{cos}\phantom{\rule{0.2em}{0ex}}u|+C

25. \int {\text{cot}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{2}{\text{cot}}^{2}u-\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sin}\phantom{\rule{0.2em}{0ex}}u|+C

26. \int {\text{sec}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}\text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}u+\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u+\text{tan}\phantom{\rule{0.2em}{0ex}}u|+C

27. \int {\text{csc}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{2}\text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cot}\phantom{\rule{0.2em}{0ex}}u+\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{csc}\phantom{\rule{0.2em}{0ex}}u-\text{cot}\phantom{\rule{0.2em}{0ex}}u|+C

28. \int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{n}{\text{sin}}^{n-1}u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+\frac{n-1}{n}\int {\text{sin}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du

29. \int {\text{cos}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n}{\text{cos}}^{n-1}u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+\frac{n-1}{n}\int {\text{cos}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du

30. \int {\text{tan}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n-1}{\text{tan}}^{n-1}u-\int {\text{tan}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du

31. \int {\text{cot}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{-1}{n-1}{\text{cot}}^{n-1}u-\int {\text{cot}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du

32. \int {\text{sec}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n-1}\text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{sec}}^{n-2}u+\frac{n-2}{n-1}\int {\text{sec}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du

33. \int {\text{csc}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{-1}{n-1}\text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{csc}}^{n-2}u+\frac{n-2}{n-1}\int {\text{csc}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du

34. \int \text{sin}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{\text{sin}\left(a-b\right)u}{2\left(a-b\right)}-\frac{\text{sin}\left(a+b\right)u}{2\left(a+b\right)}+C

35. \int \text{cos}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{\text{sin}\left(a-b\right)u}{2\left(a-b\right)}+\frac{\text{sin}\left(a+b\right)u}{2\left(a+b\right)}+C

36. \int \text{sin}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=-\frac{\text{cos}\left(a-b\right)u}{2\left(a-b\right)}-\frac{\text{cos}\left(a+b\right)u}{2\left(a+b\right)}+C

37. \int u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sin}\phantom{\rule{0.2em}{0ex}}u-u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+C

38. \int u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{cos}\phantom{\rule{0.2em}{0ex}}u+u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+C

39. \int {u}^{n}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}{u}^{n}\text{cos}\phantom{\rule{0.2em}{0ex}}u+n\int {u}^{n-1}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du

40. \int {u}^{n}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du={u}^{n}\text{sin}\phantom{\rule{0.2em}{0ex}}u-n\int {u}^{n-1}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du

41. \begin{array}{cc}\hfill \int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du& =-\frac{{\text{sin}}^{n-1}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m+1}\phantom{\rule{0.2em}{0ex}}u}{n+m}+\frac{n-1}{n+m}\int {\text{sin}}^{n-2}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m}u\phantom{\rule{0.2em}{0ex}}du\hfill \\ & =\frac{{\text{sin}}^{n+1}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m-1}u}{n+m}+\frac{m-1}{n+m}\int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m-2}u\phantom{\rule{0.2em}{0ex}}du\hfill \end{array}

Exponential and Logarithmic Integrals

42. \int u{e}^{au}\phantom{\rule{0.2em}{0ex}}du=\frac{1}{{a}^{2}}\left(au-1\right){e}^{au}+C

43. \int {u}^{n}{e}^{au}\phantom{\rule{0.2em}{0ex}}du=\frac{1}{a}{u}^{n}{e}^{au}-\frac{n}{a}\int {u}^{n-1}{e}^{au}\phantom{\rule{0.2em}{0ex}}du

44. \int {e}^{au}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}\left(a\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu-b\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\right)+C

45. \int {e}^{au}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}\left(a\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu+b\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\right)+C

46. \int \text{ln}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}u-u+C

47. \int {u}^{n}\text{ln}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{n+1}}{{\left(n+1\right)}^{2}}\left[\left(n+1\right)\text{ln}\phantom{\rule{0.1em}{0ex}}u-1\right]+C

48. \int \frac{1}{u\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}u}\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{ln}\phantom{\rule{0.1em}{0ex}}u|+C

Hyperbolic Integrals

49. \int \text{sinh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{cosh}\phantom{\rule{0.2em}{0ex}}u+C

50. \int \text{cosh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sinh}\phantom{\rule{0.2em}{0ex}}u+C

51. \int \text{tanh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}\text{cosh}\phantom{\rule{0.2em}{0ex}}u+C

52. \int \text{coth}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sinh}\phantom{\rule{0.2em}{0ex}}u|+C

53. \int \text{sech}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du={\text{tan}}^{-1}|\text{sinh}\phantom{\rule{0.2em}{0ex}}u|+C

54. \int \text{csch}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{tanh}\phantom{\rule{0.1em}{0ex}}\frac{1}{2}u|+C

55. \int {\text{sech}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tanh}\phantom{\rule{0.2em}{0ex}}u+C

56. \int {\text{csch}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{coth}\phantom{\rule{0.2em}{0ex}}u+C

57. \int \text{sech}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tanh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{sech}\phantom{\rule{0.2em}{0ex}}u+C

58. \int \text{csch}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{coth}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{csch}\phantom{\rule{0.2em}{0ex}}u+C

Inverse Trigonometric Integrals

59. \int {\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}u+\sqrt{1-{u}^{2}}+C

60. \int {\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}u-\sqrt{1-{u}^{2}}+C

61. \int {\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}u-\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+{u}^{2}\right)+C

62. \int u\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-1}{4}{\text{sin}}^{-1}u+\frac{u\sqrt{1-{u}^{2}}}{4}+C

63. \int u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-1}{4}{\text{cos}}^{-1}u-\frac{u\sqrt{1-{u}^{2}}}{4}+C

64. \int u\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{2}+1}{2}{\text{tan}}^{-1}u-\frac{u}{2}+C

65. \int {u}^{n}{\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{sin}}^{-1}u-\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{1-{u}^{2}}}\right],n\ne \text{−}1

66. \int {u}^{n}{\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{cos}}^{-1}u+\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{1-{u}^{2}}}\right],n\ne \text{−}1

67. \int {u}^{n}{\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{tan}}^{-1}u-\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{1+{u}^{2}}\right],n\ne \text{−}1

Integrals Involving a2 + u2, a > 0

68. \int \sqrt{{a}^{2}+{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}+{u}^{2}}+\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C

69. \int {u}^{2}\sqrt{{a}^{2}+{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left({a}^{2}+2{u}^{2}\right)\sqrt{{a}^{2}+{u}^{2}}-\frac{{a}^{4}}{8}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C

70. \int \frac{\sqrt{{a}^{2}+{u}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{a}^{2}+{u}^{2}}-a\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}+{u}^{2}}}{u}|+C

71. \int \frac{\sqrt{{a}^{2}+{u}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{u}+\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C

72. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}+{u}^{2}}}=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C

73. \int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}+{u}^{2}}}=\frac{u}{2}\left(\sqrt{{a}^{2}+{u}^{2}}\right)-\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C

74. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{a}^{2}+{u}^{2}}}=-\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{\sqrt{{a}^{2}+{u}^{2}}+a}{u}|+C

75. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{a}^{2}+{u}^{2}}}=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{{a}^{2}u}+C

76. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({a}^{2}+{u}^{2}\right)}^{3\text{/}2}}=\frac{u}{{a}^{2}\sqrt{{a}^{2}+{u}^{2}}}+C

Integrals Involving u2a2, a > 0

77. \int \sqrt{{u}^{2}-{a}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C

78. \int {u}^{2}\sqrt{{u}^{2}-{a}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left(2{u}^{2}-{a}^{2}\right)\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{4}}{8}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C

79. \int \frac{\sqrt{{u}^{2}-{a}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{u}^{2}-{a}^{2}}-a{\text{cos}}^{-1}\frac{a}{|u|}+C

80. \int \frac{\sqrt{{u}^{2}-{a}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{{u}^{2}-{a}^{2}}}{u}+\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C

81. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{u}^{2}-{a}^{2}}}=\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C

82. \int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{u}^{2}-{a}^{2}}}=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}+\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C

83. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{u}^{2}-{a}^{2}}}=\frac{\sqrt{{u}^{2}-{a}^{2}}}{{a}^{2}u}+C

84. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({u}^{2}-{a}^{2}\right)}^{3\text{/}2}}=\text{−}\frac{u}{{a}^{2}\sqrt{{u}^{2}-{a}^{2}}}+C

Integrals Involving a2u2, a > 0

85. \int \sqrt{{a}^{2}-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{\text{sin}}^{-1}\frac{u}{a}+C

86. \int {u}^{2}\sqrt{{a}^{2}-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left(2{u}^{2}-{a}^{2}\right)\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{4}}{8}{\text{sin}}^{-1}\frac{u}{a}+C

87. \int \frac{\sqrt{{a}^{2}-{u}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{a}^{2}-{u}^{2}}-a\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C

88. \int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{u}\sqrt{{a}^{2}-{u}^{2}}-{\text{sin}}^{-1}\frac{u}{a}+C

89. \int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}-{u}^{2}}}=-\frac{u}{u}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{\text{sin}}^{-1}\frac{u}{a}+C

90. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C

91. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{{a}^{2}u}\sqrt{{a}^{2}-{u}^{2}}+C

92. \int {\left({a}^{2}-{u}^{2}\right)}^{3\text{/}2}\phantom{\rule{0.2em}{0ex}}du=-\frac{u}{8}\left(2{u}^{2}-5{a}^{2}\right)\sqrt{{a}^{2}-{u}^{2}}+\frac{3{a}^{4}}{8}{\text{sin}}^{-1}\frac{u}{a}+C

93. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({a}^{2}-{u}^{2}\right)}^{3\text{/}2}}=-\frac{u}{{a}^{2}\sqrt{{a}^{2}-{u}^{2}}}+C

Integrals Involving 2auu2, a > 0

94. \int \sqrt{2au-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u-a}{2}\sqrt{2au-{u}^{2}}+\frac{{a}^{2}}{2}{\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C

95. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{2au-{u}^{2}}}={\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C

96. \int u\sqrt{2au-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-au-3{a}^{2}}{6}\sqrt{2au-{u}^{2}}+\frac{{a}^{3}}{2}{\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C

97. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{2au-{u}^{2}}}=-\frac{\sqrt{2au-{u}^{2}}}{au}+C

Integrals Involving a + bu, a ≠ 0

98. \int \frac{u\phantom{\rule{0.2em}{0ex}}du}{a+bu}=\frac{1}{{b}^{2}}\left(a+bu-a\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right)+C

99. \int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{a+bu}=\frac{1}{2{b}^{3}}\left[{\left(a+bu\right)}^{2}-4a\left(a+bu\right)+2{a}^{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right]+C

100. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\left(a+bu\right)}=\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{u}{a+bu}|+C

101. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\left(a+bu\right)}=-\frac{1}{au}+\frac{b}{{a}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+bu}{u}|+C

102. \int \frac{u\phantom{\rule{0.2em}{0ex}}du}{{\left(a+bu\right)}^{2}}=\frac{a}{{b}^{2}\left(a+bu\right)}+\frac{1}{{b}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|+C

103. \int \frac{u\phantom{\rule{0.2em}{0ex}}du}{u\phantom{\rule{0.2em}{0ex}}{\left(a+bu\right)}^{2}}=\frac{1}{a\left(a+bu\right)}-\frac{1}{{a}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+bu}{u}|+C

104. \int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{{\left(a+bu\right)}^{2}}=\frac{1}{{b}^{3}}\left(a+bu-\frac{{a}^{2}}{a+bu}-2a\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right)+C

105. \int u\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du=\frac{2}{15{b}^{2}}\left(3bu-2a\right){\left(a+bu\right)}^{3\text{/}2}+C

106. \int \frac{u\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2}{3{b}^{2}}\left(bu-2a\right)\sqrt{a+bu}+C

107. \int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2}{15{b}^{3}}\left(8{a}^{2}+3{b}^{2}{u}^{2}-4abu\right)\sqrt{a+bu}+C

108. \begin{array}{ccc}\hfill \int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}& =\frac{1}{\sqrt{a}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{\sqrt{a+bu}-\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}|+C,\hfill & \text{if}\phantom{\rule{0.2em}{0ex}}a>0\hfill \\ & =\frac{2}{\sqrt{\text{−}a}}\text{tan}-1\sqrt{\frac{a+bu}{\text{−}a}}+C,\hfill & \text{if}\phantom{\rule{0.2em}{0ex}}a<0\hfill \end{array}

109. \int \frac{\sqrt{a+bu}}{u}\phantom{\rule{0.2em}{0ex}}du=2\sqrt{a+bu}+a\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}

110. \int \frac{\sqrt{a+bu}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{a+bu}}{u}+\frac{b}{2}\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}

111. \int {u}^{n}\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du=\frac{2}{b\left(2n+3\right)}\left[{u}^{n}{\left(a+bu\right)}^{3\text{/}2}-na\int {u}^{n-1}\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du\right]

112. \int \frac{{u}^{n}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2{u}^{n}\sqrt{a+bu}}{b\left(2n+1\right)}-\frac{2na}{b\left(2n+1\right)}\int \frac{{u}^{n-1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}

113. \int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{n}\sqrt{a+bu}}=-\frac{\sqrt{a+bu}}{a\left(n-1\right){u}^{n-1}}-\frac{b\left(2n-3\right)}{2a\left(n-1\right)}\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{n-1}\sqrt{a+bu}}

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