Table of Integrals

Gilbert Strang; Edwin; Scott Scruggs

Basic Integrals

1. $\int {u}^{n}du=\frac{{u}^{n+1}}{n+1}+C,n\ne -1$

2. $\int \frac{du}{u}=\text{ln}|u|+C$

3. $\int {e}^{u}du={e}^{u}+C$

4. $\int {a}^{u}du=\frac{{a}^{u}}{\text{ln}a}+C$

5. $\int \sin udu=\text{-cos}u+C$

6. $\int \cos udu={\sin}u+C$

7. $\int { \sec }^{2}udu= \tan u+C$

8. $\int { \csc }^{2}udu=\text{-cot}u+C$

9. $\int \sec u \tan udu= \sec u+C$

10. $\int \csc u \cot udu=\text{-csc}u+C$

11. $\int \tan udu=\text{ln}| \sec u|+C$

12. $\int \cot udu=\text{ln}| \sin u|+C$

13. $\int \sec udu=\text{ln}| \sec u+ \tan u|+C$

14. $\int \csc udu=\text{ln}| \csc u- \cot u|+C$

15. $\int \frac{du}{\sqrt{{a}^{2}-{u}^{2}}}={ \sin }^{-1}\frac{u}{a}+C$

16. $\int \frac{du}{{a}^{2}+{u}^{2}}=\frac{1}{a}{ \tan }^{-1}\frac{u}{a}+C$

17. $\int \frac{du}{u\sqrt{{u}^{2}-{a}^{2}}}=\frac{1}{a}{ \sec }^{-1}\frac{u}{a}+C$

Trigonometric Integrals

18. $\int { \sin }^{2}udu=\frac{1}{2}u-\frac{1}{4} \sin 2u+C$

19. $\int { \cos }^{2}udu=\frac{1}{2}u+\frac{1}{4} \sin 2u+C$

20. $\int { \tan }^{2}udu= \tan u-u+C$

21. $\int { \cot }^{2}udu=- \cot u-u+C$

22. $\int { \sin }^{3}udu=-\frac{1}{3}(2+{ \sin }^{2}u) \cos u+C$

23. $\int { \cos }^{3}udu=\frac{1}{3}(2+{ \cos }^{2}u) \sin u+C$

24. $\int { \tan }^{3}udu=\frac{1}{2}{ \tan }^{2}u+\text{ln}| \cos u|+C$

25. $\int { \cot }^{3}udu=-\frac{1}{2}{ \cot }^{2}u-\text{ln}| \sin u|+C$

26. $\int { \sec }^{3}udu=\frac{1}{2} \sec u \tan u+\frac{1}{2}\text{ln}| \sec u+ \tan u|+C$

27. $\int { \csc }^{3}udu=-\frac{1}{2} \csc u \cot u+\frac{1}{2}\text{ln}| \csc u- \cot u|+C$

28. $\int { \sin }^{n}udu=-\frac{1}{n}{ \sin }^{n-1}u \cos u+\frac{n-1}{n}\int { \sin }^{n-2}udu$

29. $\int { \cos }^{n}udu=\frac{1}{n}{ \cos }^{n-1}u \sin u+\frac{n-1}{n}\int { \cos }^{n-2}udu$

30. $\int { \tan }^{n}udu=\frac{1}{n-1}{ \tan }^{n-1}u-\int { \tan }^{n-2}udu$

31. $\int { \cot }^{n}udu=\frac{-1}{n-1}{ \cot }^{n-1}u-\int { \cot }^{n-2}udu$

32. $\int { \sec }^{n}udu=\frac{1}{n-1} \tan u{ \sec }^{n-2}u+\frac{n-2}{n-1}\int { \sec }^{n-2}udu$

33. $\int { \csc }^{n}udu=\frac{-1}{n-1} \cot u{ \csc }^{n-2}u+\frac{n-2}{n-1}\int { \csc }^{n-2}udu$

34. $\int \sin au \sin budu=\frac{ \sin (a-b)u}{2(a-b)}-\frac{ \sin (a+b)u}{2(a+b)}+C$

35. $\int \cos au \cos budu=\frac{ \sin (a-b)u}{2(a-b)}+\frac{ \sin (a+b)u}{2(a+b)}+C$

36. $\int \sin au \cos budu=-\frac{ \cos (a-b)u}{2(a-b)}-\frac{ \cos (a+b)u}{2(a+b)}+C$

37. $\int u \sin udu= \sin u-u \cos u+C$

38. $\int u \cos udu= \cos u+u \sin u+C$

39. $\int {u}^{n} \sin udu=-{u}^{n} \cos u+n\int {u}^{n-1} \cos udu$

40. $\int {u}^{n} \cos udu={u}^{n} \sin u-n\int {u}^{n-1} \sin udu$

41. $\begin{array}{cc}\hfill \int { \sin }^{n}u{ \cos }^{m}udu& =-\frac{{ \sin }^{n-1}u{ \cos }^{m+1}u}{n+m}+\frac{n-1}{n+m}\int { \sin }^{n-2}u{ \cos }^{m}udu\hfill \\ & =\frac{{ \sin }^{n+1}u{ \cos }^{m-1}u}{n+m}+\frac{m-1}{n+m}\int { \sin }^{n}u{ \cos }^{m-2}udu\hfill \end{array}$

Exponential and Logarithmic Integrals

42. $\int u{e}^{au}du=\frac{1}{{a}^{2}}(au-1){e}^{au}+C$

43. $\int {u}^{n}{e}^{au}du=\frac{1}{a}{u}^{n}{e}^{au}-\frac{n}{a}\int {u}^{n-1}{e}^{au}du$

44. $\int {e}^{au} \sin budu=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \sin bu-b \cos bu)+C$

45. $\int {e}^{au} \cos budu=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \cos bu+b \sin bu)+C$

46. $\int \text{ln}udu=u\text{ln}u-u+C$

47. $\int {u}^{n}\text{ln}udu=\frac{{u}^{n+1}}{{(n+1)}^{2}}\left[(n+1)\text{ln}u-1\right]+C$

48. $\int \frac{1}{u\text{ln}u}du=\text{ln}|\text{ln}u|+C$

Hyperbolic Integrals

49. $\int \text{sinh}udu=\text{cosh}u+C$

50. $\int \text{cosh}udu=\text{sinh}u+C$

51. $\int \text{tanh}udu=\text{ln}\text{cosh}u+C$

52. $\int \text{coth}udu=\text{ln}|\text{sinh}u|+C$

53. $\int \text{sech}udu={ \tan }^{-1}|\text{sinh}u|+C$

54. $\int \text{csch}udu=\text{ln}|\text{tanh}\frac{1}{2}u|+C$

55. $\int {\text{sech}}^{2}udu=\text{tanh}u+C$

56. $\int {\text{csch}}^{2}udu=-\text{coth}u+C$

57. $\int \text{sech}u\text{tanh}udu=-\text{sech}u+C$

58. $\int \text{csch}u\text{coth}udu=-\text{csch}u+C$

Inverse Trigonometric Integrals

59. $\int { \sin }^{-1}udu=u{ \sin }^{-1}u+\sqrt{1-{u}^{2}}+C$

60. $\int { \cos }^{-1}udu=u{ \cos }^{-1}u-\sqrt{1-{u}^{2}}+C$

61. $\int { \tan }^{-1}udu=u{ \tan }^{-1}u-\frac{1}{2}\text{ln}(1+{u}^{2})+C$

62. $\int u{ \sin }^{-1}udu=\frac{2{u}^{2}-1}{4}{ \sin }^{-1}u+\frac{u\sqrt{1-{u}^{2}}}{4}+C$

63. $\int u{ \cos }^{-1}udu=\frac{2{u}^{2}-1}{4}{ \cos }^{-1}u-\frac{u\sqrt{1-{u}^{2}}}{4}+C$

64. $\int u{ \tan }^{-1}udu=\frac{{u}^{2}+1}{2}{ \tan }^{-1}u-\frac{u}{2}+C$

65. $\int {u}^{n}{ \sin }^{-1}udu=\frac{1}{n+1}\left[{u}^{n+1}{ \sin }^{-1}u-\int \frac{{u}^{n+1}du}{\sqrt{1-{u}^{2}}}\right],n\ne -1$

66. $\int {u}^{n}{ \cos }^{-1}udu=\frac{1}{n+1}\left[{u}^{n+1}{ \cos }^{-1}u+\int \frac{{u}^{n+1}du}{\sqrt{1-{u}^{2}}}\right],n\ne -1$

67. $\int {u}^{n}{ \tan }^{-1}udu=\frac{1}{n+1}\left[{u}^{n+1}{ \tan }^{-1}u-\int \frac{{u}^{n+1}du}{1+{u}^{2}}\right],n\ne -1$

Integrals Involving $a$ ² + $u$ ² , $a \symbol{"3E} 0$

68. $\int \sqrt{{a}^{2}+{u}^{2}}du=\frac{u}{2}\sqrt{{a}^{2}+{u}^{2}}+\frac{{a}^{2}}{2}\text{ln}(u+\sqrt{{a}^{2}+{u}^{2}})+C$

69. $\int {u}^{2}\sqrt{{a}^{2}+{u}^{2}}du=\frac{u}{8}({a}^{2}+2{u}^{2})\sqrt{{a}^{2}+{u}^{2}}-\frac{{a}^{4}}{8}\text{ln}(u+\sqrt{{a}^{2}+{u}^{2}})+C$

70. $\int \frac{\sqrt{{a}^{2}+{u}^{2}}}{u}du=\sqrt{{a}^{2}+{u}^{2}}-a\text{ln}|\frac{a+\sqrt{{a}^{2}+{u}^{2}}}{u}|+C$

71. $\int \frac{\sqrt{{a}^{2}+{u}^{2}}}{{u}^{2}}du=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{u}+\text{ln}(u+\sqrt{{a}^{2}+{u}^{2}})+C$

72. $\int \frac{du}{\sqrt{{a}^{2}+{u}^{2}}}=\text{ln}(u+\sqrt{{a}^{2}+{u}^{2}})+C$

73. $\int \frac{{u}^{2}du}{\sqrt{{a}^{2}+{u}^{2}}}=\frac{u}{2}(\sqrt{{a}^{2}+{u}^{2}})-\frac{{a}^{2}}{2}\text{ln}(u+\sqrt{{a}^{2}+{u}^{2}})+C$

74. $\int \frac{du}{u\sqrt{{a}^{2}+{u}^{2}}}=-\frac{1}{a}\text{ln}|\frac{\sqrt{{a}^{2}+{u}^{2}}+a}{u}|+C$

75. $\int \frac{du}{{u}^{2}\sqrt{{a}^{2}+{u}^{2}}}=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{{a}^{2}u}+C$

76. $\int \frac{du}{{({a}^{2}+{u}^{2})}^{3\text{/}2}}=\frac{u}{{a}^{2}\sqrt{{a}^{2}+{u}^{2}}}+C$

Integrals Involving $u$ ² – $a$ ² , $a \symbol{"3E} 0$

77. $\int \sqrt{{u}^{2}-{a}^{2}}du=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{2}}{2}\text{ln}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

78. $\int {u}^{2}\sqrt{{u}^{2}-{a}^{2}}du=\frac{u}{8}(2{u}^{2}-{a}^{2})\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{4}}{8}\text{ln}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

79. $\int \frac{\sqrt{{u}^{2}-{a}^{2}}}{u}du=\sqrt{{u}^{2}-{a}^{2}}-a{ \cos }^{-1}\frac{a}{|u|}+C$

80. $\int \frac{\sqrt{{u}^{2}-{a}^{2}}}{{u}^{2}}du=-\frac{\sqrt{{u}^{2}-{a}^{2}}}{u}+\text{ln}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

81. $\int \frac{du}{\sqrt{{u}^{2}-{a}^{2}}}=\text{ln}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

82. $\int \frac{{u}^{2}du}{\sqrt{{u}^{2}-{a}^{2}}}=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}+\frac{{a}^{2}}{2}\text{ln}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

83. $\int \frac{du}{{u}^{2}\sqrt{{u}^{2}-{a}^{2}}}=\frac{\sqrt{{u}^{2}-{a}^{2}}}{{a}^{2}u}+C$

84. $\int \frac{du}{{({u}^{2}-{a}^{2})}^{3\text{/}2}}=-\frac{u}{{a}^{2}\sqrt{{u}^{2}-{a}^{2}}}+C$

Integrals Involving $a$ ² – $u$ ² , $a \symbol{"3E}$

85. $\int \sqrt{{a}^{2}-{u}^{2}}du=\frac{u}{2}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{ \sin }^{-1}\frac{u}{a}+C$

86. $\int {u}^{2}\sqrt{{a}^{2}-{u}^{2}}du=\frac{u}{8}(2{u}^{2}-{a}^{2})\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{4}}{8}{ \sin }^{-1}\frac{u}{a}+C$

87. $\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{u}du=\sqrt{{a}^{2}-{u}^{2}}-a\text{ln}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C$

88. $\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\frac{1}{u}\sqrt{{a}^{2}-{u}^{2}}-{ \sin }^{-1}\frac{u}{a}+C$

89. $\int \frac{{u}^{2}du}{\sqrt{{a}^{2}-{u}^{2}}}=-\frac{u}{u}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{ \sin }^{-1}\frac{u}{a}+C$

90. $\int \frac{du}{u\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{a}\text{ln}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C$

91. $\int \frac{du}{{u}^{2}\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{{a}^{2}u}\sqrt{{a}^{2}-{u}^{2}}+C$

92. $\int {({a}^{2}-{u}^{2})}^{3\text{/}2}du=-\frac{u}{8}(2{u}^{2}-5{a}^{2})\sqrt{{a}^{2}-{u}^{2}}+\frac{3{a}^{4}}{8}{ \sin }^{-1}\frac{u}{a}+C$

93. $\int \frac{du}{{({a}^{2}-{u}^{2})}^{3\text{/}2}}=-\frac{u}{{a}^{2}\sqrt{{a}^{2}-{u}^{2}}}+C$

Integrals Involving 2 au – $u$ ² , $a \symbol{"3E} 0$

94. $\int \sqrt{2au-{u}^{2}}du=\frac{u-a}{2}\sqrt{2au-{u}^{2}}+\frac{{a}^{2}}{2}{ \cos }^{-1}(\frac{a-u}{a})+C$

95. $\int \frac{du}{\sqrt{2au-{u}^{2}}}={ \cos }^{-1}(\frac{a-u}{a})+C$

96. $\int u\sqrt{2au-{u}^{2}}du=\frac{2{u}^{2}-au-3{a}^{2}}{6}\sqrt{2au-{u}^{2}}+\frac{{a}^{3}}{2}{ \cos }^{-1}(\frac{a-u}{a})+C$

97. $\int \frac{du}{u\sqrt{2au-{u}^{2}}}=-\frac{\sqrt{2au-{u}^{2}}}{au}+C$

Integrals Involving $a$ + bu , $a$ ≠ 0

98. $\int \frac{udu}{a+bu}=\frac{1}{{b}^{2}}(a+bu-a\text{ln}|a+bu|)+C$

99. $\int \frac{{u}^{2}du}{a+bu}=\frac{1}{2{b}^{3}}\left[{(a+bu)}^{2}-4a(a+bu)+2{a}^{2}\text{ln}|a+bu|\right]+C$

100. $\int \frac{du}{u(a+bu)}=\frac{1}{a}\text{ln}|\frac{u}{a+bu}|+C$

101. $\int \frac{du}{{u}^{2}(a+bu)}=-\frac{1}{au}+\frac{b}{{a}^{2}}\text{ln}|\frac{a+bu}{u}|+C$

102. $\int \frac{udu}{{(a+bu)}^{2}}=\frac{a}{{b}^{2}(a+bu)}+\frac{1}{{b}^{2}}\text{ln}|a+bu|+C$

103. $\int \frac{udu}{u{(a+bu)}^{2}}=\frac{1}{a(a+bu)}-\frac{1}{{a}^{2}}\text{ln}|\frac{a+bu}{u}|+C$

104. $\int \frac{{u}^{2}du}{{(a+bu)}^{2}}=\frac{1}{{b}^{3}}(a+bu-\frac{{a}^{2}}{a+bu}-2a\text{ln}|a+bu|)+C$

105. $\int u\sqrt{a+bu}du=\frac{2}{15{b}^{2}}(3bu-2a){(a+bu)}^{3\text{/}2}+C$

106. $\int \frac{udu}{\sqrt{a+bu}}=\frac{2}{3{b}^{2}}(bu-2a)\sqrt{a+bu}+C$

107. $\int \frac{{u}^{2}du}{\sqrt{a+bu}}=\frac{2}{15{b}^{3}}(8{a}^{2}+3{b}^{2}{u}^{2}-4abu)\sqrt{a+bu}+C$

108. $\begin{array}{ccc}\hfill \int \frac{du}{u\sqrt{a+bu}}& =\frac{1}{\sqrt{a}}\text{ln}|\frac{\sqrt{a+bu}-\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}|+C,\hfill & \text{ if }a \symbol{"3E} 0\hfill \\ & =\frac{2}{\sqrt{-a}} \tan -1\sqrt{\frac{a+bu}{-a}}+C,\hfill & \text{ if }a<0\hfill \end{array}$