Basic Integrals

1. [latex]\displaystyle\int {u}^{n}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1[/latex]

2. [latex]\displaystyle\int \frac{du}{u}=\text{ln}|u|+C[/latex]

3. [latex]\displaystyle\int {e}^{u}du={e}^{u}+C[/latex]

4. [latex]\displaystyle\int {a}^{u}du=\frac{{a}^{u}}{\text{ln}a}+C[/latex]

5. [latex]\displaystyle\int \sin udu=\text{−cos}u+C[/latex]

6. [latex]\displaystyle\int \cos udu={\sin}u+C[/latex]

7. [latex]\displaystyle\int { \sec }^{2}udu= \tan u+C[/latex]

8. [latex]\displaystyle\int { \csc }^{2}udu=\text{−cot}u+C[/latex]

9. [latex]\displaystyle\int \sec u \tan udu= \sec u+C[/latex]

10. [latex]\displaystyle\int \csc u \cot udu=\text{−csc}u+C[/latex]

11. [latex]\displaystyle\int \tan udu=\text{ln}| \sec u|+C[/latex]

12. [latex]\displaystyle\int \cot udu=\text{ln}| \sin u|+C[/latex]

13. [latex]\displaystyle\int \sec udu=\text{ln}| \sec u+ \tan u|+C[/latex]

14. [latex]\displaystyle\int \csc udu=\text{ln}| \csc u- \cot u|+C[/latex]

15. [latex]\displaystyle\int \frac{du}{\sqrt{{a}^{2}-{u}^{2}}}={ \sin }^{-1}\frac{u}{a}+C[/latex]

16. [latex]\displaystyle\int \frac{du}{{a}^{2}+{u}^{2}}=\frac{1}{a}{ \tan }^{-1}\frac{u}{a}+C[/latex]

17. [latex]\displaystyle\int \frac{du}{u\sqrt{{u}^{2}-{a}^{2}}}=\frac{1}{a}{ \sec }^{-1}\frac{u}{a}+C[/latex]

Trigonometric Integrals

18. [latex]\displaystyle\int { \sin }^{2}udu=\frac{1}{2}u-\frac{1}{4} \sin 2u+C[/latex]

19. [latex]\displaystyle\int { \cos }^{2}udu=\frac{1}{2}u+\frac{1}{4} \sin 2u+C[/latex]

20. [latex]\displaystyle\int { \tan }^{2}udu= \tan u-u+C[/latex]

21. [latex]\displaystyle\int { \cot }^{2}udu=\text{−} \cot u-u+C[/latex]

22. [latex]\displaystyle\int { \sin }^{3}udu=-\frac{1}{3}(2+{ \sin }^{2}u) \cos u+C[/latex]

23. [latex]\displaystyle\int { \cos }^{3}udu=\frac{1}{3}(2+{ \cos }^{2}u) \sin u+C[/latex]

24. [latex]\displaystyle\int { \tan }^{3}udu=\frac{1}{2}{ \tan }^{2}u+\text{ln}| \cos u|+C[/latex]

25. [latex]\displaystyle\int { \cot }^{3}udu=-\frac{1}{2}{ \cot }^{2}u-\text{ln}| \sin u|+C[/latex]

26. [latex]\displaystyle\int { \sec }^{3}udu=\frac{1}{2} \sec u \tan u+\frac{1}{2}\text{ln}| \sec u+ \tan u|+C[/latex]

27. [latex]\displaystyle\int { \csc }^{3}udu=-\frac{1}{2} \csc u \cot u+\frac{1}{2}\text{ln}| \csc u- \cot u|+C[/latex]

28. [latex]\displaystyle\int { \sin }^{n}udu=-\frac{1}{n}{ \sin }^{n-1}u \cos u+\frac{n-1}{n}\displaystyle\int { \sin }^{n-2}udu[/latex]

29. [latex]\displaystyle\int { \cos }^{n}udu=\frac{1}{n}{ \cos }^{n-1}u \sin u+\frac{n-1}{n}\displaystyle\int { \cos }^{n-2}udu[/latex]

30. [latex]\displaystyle\int { \tan }^{n}udu=\frac{1}{n-1}{ \tan }^{n-1}u-\displaystyle\int { \tan }^{n-2}udu[/latex]

31. [latex]\displaystyle\int { \cot }^{n}udu=\frac{-1}{n-1}{ \cot }^{n-1}u-\displaystyle\int { \cot }^{n-2}udu[/latex]

32. [latex]\displaystyle\int { \sec }^{n}udu=\frac{1}{n-1} \tan u{ \sec }^{n-2}u+\frac{n-2}{n-1}\displaystyle\int { \sec }^{n-2}udu[/latex]

33. [latex]\displaystyle\int { \csc }^{n}udu=\frac{-1}{n-1} \cot u{ \csc }^{n-2}u+\frac{n-2}{n-1}\displaystyle\int { \csc }^{n-2}udu[/latex]

34. [latex]\displaystyle\int \sin au \sin budu=\frac{ \sin (a-b)u}{2(a-b)}-\frac{ \sin (a+b)u}{2(a+b)}+C[/latex]

35. [latex]\displaystyle\int \cos au \cos budu=\frac{ \sin (a-b)u}{2(a-b)}+\frac{ \sin (a+b)u}{2(a+b)}+C[/latex]

36. [latex]\displaystyle\int \sin au \cos budu=-\frac{ \cos (a-b)u}{2(a-b)}-\frac{ \cos (a+b)u}{2(a+b)}+C[/latex]

37. [latex]\displaystyle\int u \sin udu= \sin u-u \cos u+C[/latex]

38. [latex]\displaystyle\int u \cos udu= \cos u+u \sin u+C[/latex]

39. [latex]\displaystyle\int {u}^{n} \sin udu=\text{−}{u}^{n} \cos u+n\displaystyle\int {u}^{n-1} \cos udu[/latex]

40. [latex]\displaystyle\int {u}^{n} \cos udu={u}^{n} \sin u-n\displaystyle\int {u}^{n-1} \sin udu[/latex]

41. [latex]\begin{array}{cc}\hfill {\displaystyle\int { \sin }^{n}u{ \cos }^{m}udu}& =-\dfrac{{ \sin }^{n-1}u{ \cos }^{m+1}u}{n+m}+\dfrac{n-1}{n+m}{\displaystyle\int { \sin }^{n-2}u{ \cos }^{m}udu}\hfill \\ & =\dfrac{{ \sin }^{n+1}u { \cos }^{m-1}u} {n+m}+\dfrac{m-1}{n+m}{\displaystyle\int { \sin }^{n}u{ \cos }^{m-2}udu}\hfill \end{array}[/latex]

Exponential and Logarithmic Integrals

42. [latex]\displaystyle\int u{e}^{au}du=\frac{1}{{a}^{2}}(au-1){e}^{au}+C[/latex]

43. [latex]\displaystyle\int {u}^{n}{e}^{au}du=\frac{1}{a}{u}^{n}{e}^{au}-\frac{n}{a}\displaystyle\int {u}^{n-1}{e}^{au}du[/latex]

44. [latex]\displaystyle\int {e}^{au} \sin budu=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \sin bu-b \cos bu)+C[/latex]

45. [latex]\displaystyle\int {e}^{au} \cos budu=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \cos bu+b \sin bu)+C[/latex]

46. [latex]\displaystyle\int \text{ln}udu=u\text{ln}u-u+C[/latex]

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

48. [latex]\displaystyle\int \frac{1}{u\text{ln}u}du=\text{ln}|\text{ln}u|+C[/latex]

Hyperbolic Integrals

49. [latex]\displaystyle\int \text{sinh}udu=\text{cosh}u+C[/latex]

50. [latex]\displaystyle\int \text{cosh}udu=\text{sinh}u+C[/latex]

51. [latex]\displaystyle\int \text{tanh}udu=\text{ln}\text{cosh}u+C[/latex]

52. [latex]\displaystyle\int \text{coth}udu=\text{ln}|\text{sinh}u|+C[/latex]

53. [latex]\displaystyle\int \text{sech}udu={ \tan }^{-1}|\text{sinh}u|+C[/latex]

54. [latex]\displaystyle\int \text{csch}udu=\text{ln}|\text{tanh}\frac{1}{2}u|+C[/latex]

55. [latex]\displaystyle\int {\text{sech}}^{2}udu=\text{tanh}u+C[/latex]

56. [latex]\displaystyle\int {\text{csch}}^{2}udu=\text{−}\text{coth}u+C[/latex]

57. [latex]\displaystyle\int \text{sech}u\text{tanh}udu=\text{−}\text{sech}u+C[/latex]

58. [latex]\displaystyle\int \text{csch}u\text{coth}udu=\text{−}\text{csch}u+C[/latex]

Inverse Trigonometric Integrals

59. [latex]\displaystyle\int { \sin }^{-1}udu=u{ \sin }^{-1}u+\sqrt{1-{u}^{2}}+C[/latex]

60. [latex]\displaystyle\int { \cos }^{-1}udu=u{ \cos }^{-1}u-\sqrt{1-{u}^{2}}+C[/latex]

61. [latex]\displaystyle\int { \tan }^{-1}udu=u{ \tan }^{-1}u-\frac{1}{2}\text{ln}(1+{u}^{2})+C[/latex]

62. [latex]\displaystyle\int u{ \sin }^{-1}udu=\frac{2{u}^{2}-1}{4}{ \sin }^{-1}u+\frac{u\sqrt{1-{u}^{2}}}{4}+C[/latex]

63. [latex]\displaystyle\int u{ \cos }^{-1}udu=\frac{2{u}^{2}-1}{4}{ \cos }^{-1}u-\frac{u\sqrt{1-{u}^{2}}}{4}+C[/latex]

64. [latex]\displaystyle\int u{ \tan }^{-1}udu=\frac{{u}^{2}+1}{2}{ \tan }^{-1}u-\frac{u}{2}+C[/latex]

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

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

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

Integrals Involving [latex]a[/latex]² + [latex]u[/latex]², [latex]a[/latex] > 0

68. [latex]\displaystyle\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[/latex]

69. [latex]\displaystyle\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[/latex]

70. [latex]\displaystyle\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[/latex]

71. [latex]\displaystyle\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[/latex]

72. [latex]\displaystyle\int \frac{du}{\sqrt{{a}^{2}+{u}^{2}}}=\text{ln}(u+\sqrt{{a}^{2}+{u}^{2}})+C[/latex]

73. [latex]\displaystyle\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[/latex]

74. [latex]\displaystyle\int \frac{du}{u\sqrt{{a}^{2}+{u}^{2}}}=-\frac{1}{a}\text{ln}|\frac{\sqrt{{a}^{2}+{u}^{2}}+a}{u}|+C[/latex]

75. [latex]\displaystyle\int \frac{du}{{u}^{2}\sqrt{{a}^{2}+{u}^{2}}}=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{{a}^{2}u}+C[/latex]

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

Integrals Involving [latex]u[/latex]² − [latex]a[/latex]², [latex]a[/latex] > 0

77. [latex]\displaystyle\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[/latex]

78. [latex]\displaystyle\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[/latex]

79. [latex]\displaystyle\int \frac{\sqrt{{u}^{2}-{a}^{2}}}{u}du=\sqrt{{u}^{2}-{a}^{2}}-a{ \cos }^{-1}\frac{a}{|u|}+C[/latex]

80. [latex]\displaystyle\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[/latex]

81. [latex]\displaystyle\int \frac{du}{\sqrt{{u}^{2}-{a}^{2}}}=\text{ln}|u+\sqrt{{u}^{2}-{a}^{2}}|+C[/latex]

82. [latex]\displaystyle\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[/latex]

83. [latex]\displaystyle\int \frac{du}{{u}^{2}\sqrt{{u}^{2}-{a}^{2}}}=\frac{\sqrt{{u}^{2}-{a}^{2}}}{{a}^{2}u}+C[/latex]

84. [latex]\displaystyle\int \frac{du}{{({u}^{2}-{a}^{2})}^{3\text{/}2}}=\text{−}\frac{u}{{a}^{2}\sqrt{{u}^{2}-{a}^{2}}}+C[/latex]

Integrals Involving [latex]a[/latex]² − [latex]u[/latex]², [latex]a[/latex] > 0

85. [latex]\displaystyle\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[/latex]

86. [latex]\displaystyle\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[/latex]

87. [latex]\displaystyle\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[/latex]

88. [latex]\displaystyle\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\frac{1}{u}\sqrt{{a}^{2}-{u}^{2}}-{ \sin }^{-1}\frac{u}{a}+C[/latex]

89. [latex]\displaystyle\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[/latex]

90. [latex]\displaystyle\int \frac{du}{u\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{a}\text{ln}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C[/latex]

91. [latex]\displaystyle\int \frac{du}{{u}^{2}\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{{a}^{2}u}\sqrt{{a}^{2}-{u}^{2}}+C[/latex]

92. [latex]\displaystyle\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[/latex]

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

Integrals Involving 2au − [latex]u[/latex]², [latex]a[/latex] > 0

94. [latex]\displaystyle\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[/latex]

95. [latex]\displaystyle\int \frac{du}{\sqrt{2au-{u}^{2}}}={ \cos }^{-1}(\frac{a-u}{a})+C[/latex]

96. [latex]\displaystyle\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[/latex]

97. [latex]\displaystyle\int \frac{du}{u\sqrt{2au-{u}^{2}}}=-\frac{\sqrt{2au-{u}^{2}}}{au}+C[/latex]

Integrals Involving [latex]a[/latex] + bu, [latex]a[/latex] ≠ 0

98. [latex]\displaystyle\int \frac{udu}{a+bu}=\frac{1}{{b}^{2}}(a+bu-a\text{ln}|a+bu|)+C[/latex]

99. [latex]\displaystyle\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[/latex]

100. [latex]\displaystyle\int \frac{du}{u(a+bu)}=\frac{1}{a}\text{ln}|\frac{u}{a+bu}|+C[/latex]

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

102. [latex]\displaystyle\int \frac{udu}{{(a+bu)}^{2}}=\frac{a}{{b}^{2}(a+bu)}+\frac{1}{{b}^{2}}\text{ln}|a+bu|+C[/latex]

103. [latex]\displaystyle\int \frac{udu}{u{(a+bu)}^{2}}=\frac{1}{a(a+bu)}-\frac{1}{{a}^{2}}\text{ln}|\frac{a+bu}{u}|+C[/latex]

104. [latex]\displaystyle\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[/latex]

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

106. [latex]\displaystyle\int \frac{udu}{\sqrt{a+bu}}=\frac{2}{3{b}^{2}}(bu-2a)\sqrt{a+bu}+C[/latex]

107. [latex]\displaystyle\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[/latex]

108. [latex]\begin{array}{ccc}\hfill \displaystyle\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>0\hfill \\ & =\frac{2}{\sqrt{\text{−}a}} \tan -1\sqrt{\frac{a+bu}{\text{−}a}}+C,\hfill & \text{ if }a<0\hfill \end{array}[/latex]

109. [latex]\displaystyle\int \frac{\sqrt{a+bu}}{u}du=2\sqrt{a+bu}+a\displaystyle\int \frac{du}{u\sqrt{a+bu}}[/latex]

110. [latex]\displaystyle\int \frac{\sqrt{a+bu}}{{u}^{2}}du=-\frac{\sqrt{a+bu}}{u}+\frac{b}{2}\displaystyle\int \frac{du}{u\sqrt{a+bu}}[/latex]

111. [latex]\displaystyle\int {u}^{n}\sqrt{a+bu}du=\frac{2}{b(2n+3)}\left[{u}^{n}{(a+bu)}^{3\text{/}2}-na\displaystyle\int {u}^{n-1}\sqrt{a+bu}du\right][/latex]

112. [latex]\displaystyle\int \frac{{u}^{n}du}{\sqrt{a+bu}}=\frac{2{u}^{n}\sqrt{a+bu}}{b(2n+1)}-\frac{2na}{b(2n+1)}\displaystyle\int \frac{{u}^{n-1}du}{\sqrt{a+bu}}[/latex]

113. [latex]\displaystyle\int \frac{du}{{u}^{n}\sqrt{a+bu}}=-\frac{\sqrt{a+bu}}{a(n-1){u}^{n-1}}-\frac{b(2n-3)}{2a(n-1)}\displaystyle\int \frac{du}{{u}^{n-1}\sqrt{a+bu}}[/latex]