# 14.8.1: Table of Integrals


This table of integrals is from another open textbook in Libretexts and OpenStax.

## Most Common Integrals

### Basic Integrals

1. $$\quad \displaystyle ∫x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n≠−1$$

2. $$\quad \displaystyle ∫\frac{dx}{x} =\ln |x|+C$$

3. $$\quad \displaystyle ∫e^x\,dx=e^x+C$$

or more generally

3. $$\quad \displaystyle ∫e^{ax}\,dx=\frac{1}{a} e^{ax}+C$$

4. $$\quad \displaystyle ∫a^x\,dx=\frac{a^x}{\ln a}+C$$

5. $$\quad \displaystyle ∫\sin x\,dx=−\cos x+C$$

6. $$\quad \displaystyle ∫\cos x\,dx=\sin x+C$$

7. $$\quad \displaystyle ∫\sec^2x\,dx=\tan x+C$$

8. $$\quad \displaystyle ∫\csc^2x\,dx=−\cot x+C$$

9. $$\quad \displaystyle ∫\sec x\tan x\,dx=\sec x+C$$

10. $$\quad \displaystyle ∫\csc x\cot x\,dx=−\csc x+C$$

11. $$\quad \displaystyle ∫\tan x\,dx=\ln |\sec x|+C$$

12. $$\quad \displaystyle ∫\cot x\,dx=\ln |\sin x|+C$$

13. $$\quad \displaystyle ∫\sec x\,dx=\ln |\sec x+\tan x|+C$$

14. $$\quad \displaystyle ∫\csc x\,dx=\ln |\csc x−\cot x|+C$$

15. $$\quad \displaystyle ∫\frac{dx}{\sqrt{a^2−x^2}}=\arcsin \left(\frac{x}{a}\right)+C$$

16. $$\quad \displaystyle ∫\frac{dx}{a^2+x^2}=\frac{1}{a}\arctan \left(\frac{x}{a}\right)+C$$

17. $$\quad \displaystyle ∫\frac{dx}{x\sqrt{x^2−a^2}}=\frac{1}{a}\text{arcsec} \left(\frac{|x|}{a}\right)+C$$

### Trigonometric Integrals

18. $$\quad \displaystyle ∫\sin^2x\,dx=\frac{1}{2}x−\frac{1}{4}\sin 2x+C$$

19. $$\quad \displaystyle ∫\cos^2 x\,dx=\frac{1}{2}x+\frac{1}{4}\sin 2x+C$$

20. $$\quad \displaystyle ∫\tan^2 x\,dx=\tan x−x+C$$

21. $$\quad \displaystyle ∫\cot ^2 x\,dx=−\cot x−x+C$$

22. $$\quad \displaystyle ∫\sin^3 x\,dx=−\frac{1}{3}(2+\sin^2x)\cos x+C$$

23. $$\quad \displaystyle ∫\cos^3 x\,dx=\frac{1}{3}(2+\cos^2 x)\sin x+C$$

24. $$\quad \displaystyle ∫\tan^3 x\,dx=\frac{1}{2}\tan^2 x+\ln |\cos x|+C$$

25. $$\quad \displaystyle ∫\cot^3 x\,dx=−\frac{1}{2}\cot^2 x−\ln |\sin x|+C$$

26. $$\quad \displaystyle ∫\sec^3 x\,dx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln |\sec x+\tan x|+C$$

27. $$\quad \displaystyle ∫\csc^3 x\,dx=−\frac{1}{2}\csc x\cot x+\frac{1}{2}\ln |\csc x−\cot x|+C$$

28. $$\quad \displaystyle ∫\sin^n x\,dx=\frac{-1}{n}\sin^{n−1}x\cos x+\frac{n−1}{n}∫\sin^{n−2}x\,dx$$

29. $$\quad \displaystyle ∫\cos^n x\,dx=\frac{1}{n}\cos^{n−1} x\sin x+\frac{n−1}{n}∫\cos^{n−2}x\,dx$$

30. $$\quad \displaystyle ∫\tan^n x\,dx=\frac{1}{n-1}\tan^{n−1} x−∫\tan^{n−2} x\,dx$$

31. $$\quad \displaystyle ∫\cot^n x\,dx=\frac{-1}{n-1}\cot^{n−1}x−∫\cot^{n−2}x\,dx$$

32. $$\quad \displaystyle ∫\sec^n x\,dx=\frac{1}{n-1}\tan x\sec^{n−2}x+\frac{n-2}{n-1}∫\sec^{n−2}x\,dx$$

33. $$\quad \displaystyle ∫\csc^n x\,dx=\frac{-1}{n-1}\cot x\csc^{n−2}x+\frac{n-2}{n-1}∫\csc^{n−2}x\,dx$$

34. $$\quad \displaystyle ∫\sin ax\sin bx\,dx=\frac{\sin (a−b)x}{2(a−b)}−\frac{\sin (a+b)x}{2(a+b)}+C$$

35. $$\quad \displaystyle ∫\cos ax\cos bx\,dx=\frac{\sin (a−b)x}{2(a−b)}+\frac{\sin (a+b)x}{2(a+b)}+C$$

36. $$\quad \displaystyle ∫\sin ax\cos bx\,dx=−\frac{\cos (a−b)x}{2(a−b)}−\frac{\cos (a+b)x}{2(a+b)}+C$$

37. $$\quad \displaystyle ∫x\sin x\,dx=\sin x−x\cos x+C$$

38. $$\quad \displaystyle ∫x\cos x\,dx=\cos x+x\sin x+C$$

39. $$\quad \displaystyle ∫x^n\sin x\,dx=−x^n\cos x+n∫x^{n−1}\cos x\,dx$$

40. $$\quad \displaystyle ∫x^n\cos x\,dx=x^n\sin x−n∫x^{n−1}\sin x\,dx$$

41. $$\quad \displaystyle ∫\sin^n x\cos^m x\,dx=$$ use the methods for powers of sine and cosine

### Exponential and Logarithmic Integrals

42. $$\quad \displaystyle ∫xe^{ax}\,dx=\frac{1}{a^2}(ax−1)e^{ax}+C$$

#### Example of integration by parts

$$\qquad \int{f(x)g'(x)dx} = f(x)g(x) - \int{f'(x)g(x) dx}$$

43. $$\quad \displaystyle ∫x^ne^{ax}\,dx=\frac{1}{a}x^ne^{ax}−\frac{n}{a}∫x^{n−1}e^{ax}\,dx$$

44. $$\quad \displaystyle ∫e^{ax}\sin bx\,dx=\frac{e^{ax}}{a^2+b^2}(a\sin bx−b\cos bx)+C$$

45. $$\quad \displaystyle ∫e^{ax}\cos bx\,dx=\frac{e^{ax}}{a^2+b^2}(a\cos bx+b\sin bx)+C$$

46. $$\quad \displaystyle ∫\ln x\,dx=x\ln x−x+C$$

47. $$\quad \displaystyle ∫x^n\ln x\,dx=\frac{x^{n+1}}{(n+1)^2}[(n+1)\ln x−1]+C$$

48. $$\quad \displaystyle ∫\frac{1}{x\ln x}\,dx=\ln |\ln x|+C$$

## Less Common Integrals

### Hyperbolic Integrals

49. $$\quad \displaystyle ∫\sinh x\,dx=\cosh x+C$$

50. $$\quad \displaystyle ∫\cosh x\,dx=\sinh x+C$$

51. $$\quad \displaystyle ∫\tanh x\,dx=\ln \cosh x+C$$

52. $$\quad \displaystyle ∫\coth x\,dx=\ln |\sinh x|+C$$

53. $$\quad \displaystyle ∫\text{sech}\,x\,dx=\arctan |\sinh x|+C$$

54. $$\quad \displaystyle ∫\text{csch}\,x\,dx=\ln ∣\tanh\frac{1}{2}x∣+C$$

55. $$\quad \displaystyle ∫\text{sech}^2 x\,dx=\tanh \,x+C$$

56. $$\quad \displaystyle ∫\text{csch}^2 x\,dx=−\coth \,x+C$$

57. $$\quad \displaystyle ∫\text{sech} \,x\tanh x\,dx=−\text{sech} \,x+C$$

58. $$\quad \displaystyle ∫\text{csch} \,x\coth x\,dx=−\text{csch} \,x+C$$

### Inverse Trigonometric Integrals

59. $$\quad \displaystyle ∫\arcsin x\,dx=x\arcsin x+\sqrt{1−x^2}+C$$

60. $$\quad \displaystyle ∫\arccos x\,dx=x\arccos x−\sqrt{1−x^2}+C$$

61. $$\quad \displaystyle ∫\arctan x\,dx=x\arctan x−\frac{1}{2}\ln (1+x^2)+C$$

62. $$\quad \displaystyle ∫x\arcsin x\,dx=\frac{2x^2−1}{4}\arcsin x+\frac{x\sqrt{1−x^2}}{4}+C$$

63. $$\quad \displaystyle ∫x\arccos x\,dx=\frac{2x^2−1}{4}\arccos x-\frac{x\sqrt{1−x^2}}{4}+C$$

64. $$\quad \displaystyle ∫x\arctan x\,dx=\frac{x^2+1}{2}\arctan x−\frac{x}{2}+C$$

65. $$\quad \displaystyle ∫x^n\arcsin x\,dx=\frac{1}{n+1}\left[x^{n+1}\arcsin x−∫\frac{x^{n+1}\,dx}{\sqrt{1−x^2}}\right],\quad n≠−1$$

66. $$\quad \displaystyle ∫x^n\arccos x\,dx=\frac{1}{n+1}\left[x^{n+1}\arccos x+∫\frac{x^{n+1}\,dx}{\sqrt{1−x^2}}\right],\quad n≠−1$$

67. $$\quad \displaystyle ∫x^n\arctan x\,dx=\frac{1}{n+1}\left[x^{n+1}\arctan x−∫\frac{x^{n+1}\,dx}{1+x^2}\right],\quad n≠−1$$

## Integrals with Fractions and Square roots

### Integrals Involving a2 + x2, a > 0

68. $$\quad \displaystyle ∫\sqrt{a^2+x^2}\,dx=\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\ln \left(x+\sqrt{a^2+x^2}\right)+C$$

69. $$\quad \displaystyle ∫x^2\sqrt{a^2+x^2}\,dx=\frac{x}{8}(a^2+2x^2)\sqrt{a^2+x^2}−\frac{a^4}{8}\ln \left(x+\sqrt{a^2+x^2}\right)+C$$

70. $$\quad \displaystyle ∫\frac{\sqrt{a^2+x^2}}{x}\,dx=\sqrt{a^2+x^2}−a\ln \left|\frac{a+\sqrt{a^2+x^2}}{x}\right|+C$$

71. $$\quad \displaystyle ∫\frac{\sqrt{a^2+x^2}}{x^2}\,dx=−\frac{\sqrt{a^2+x^2}}{x}+\ln \left(x+\sqrt{a^2+x^2}\right)+C$$

72. $$\quad \displaystyle ∫\frac{dx}{\sqrt{a^2+x^2}}=\ln \left(x+\sqrt{a^2+x^2}\right)+C$$

73. $$\quad \displaystyle ∫\frac{x^2}{\sqrt{a^2+x^2}}\,dx=\frac{x}{2}\left(\sqrt{a^2+x^2}\right)−\frac{a^2}{2}\ln \left(x+\sqrt{a^2+x^2}\right)+C$$

74. $$\quad \displaystyle ∫\frac{dx}{x\sqrt{a^2+x^2}}=\frac{−1}{a}\ln \left|\frac{\sqrt{a^2+x^2}+a}{x}\right|+C$$

75. $$\quad \displaystyle ∫\frac{dx}{x^2\sqrt{a^2+x^2}}=−\frac{\sqrt{a^2+x^2}}{a^2x}+C$$

76. $$\quad \displaystyle ∫\frac{dx}{\left(a^2+x^2\right)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}}+C$$

### Integrals Involving x2 − a2, a > 0

77. $$\quad \displaystyle ∫\sqrt{x^2−a^2}\,dx=\frac{x}{2}\sqrt{x^2−a^2}−\frac{a^2}{2}\ln \left|x+\sqrt{x^2−a^2}\right|+C$$

78. $$\quad \displaystyle ∫x^2\sqrt{x^2−a^2}\,dx=\frac{x}{8}(2x^2−a^2)\sqrt{x^2−a^2}−\frac{a^4}{8}\ln \left|x+\sqrt{x^2−a^2}\right|+C$$

79. $$\quad \displaystyle ∫\frac{\sqrt{x^2−a^2}}{x}\,dx=\sqrt{x^2−a^2}−a\arccos\frac{a}{|x|}+C$$

80. $$\quad \displaystyle ∫\frac{\sqrt{x^2−a^2}}{x^2}\,dx=−\frac{\sqrt{x^2−a^2}}{x}+\ln \left|x+\sqrt{x^2−a^2}\right|+C$$

81. $$\quad \displaystyle ∫\frac{dx}{\sqrt{x^2−a^2}}=\ln \left|x+\sqrt{x^2−a^2}\right|+C$$

82. $$\quad \displaystyle ∫\frac{x^2}{\sqrt{x^2−a^2}}\,dx=\frac{x}{2}\sqrt{x^2−a^2}+\frac{a^2}{2}\ln \left|x+\sqrt{x^2−a^2}\right|+C$$

83. $$\quad \displaystyle ∫\frac{dx}{x^2\sqrt{x^2−a^2}}=\frac{\sqrt{x^2−a^2}}{a^2x}+C$$

84. $$\quad \displaystyle ∫\frac{dx}{(x^2−a^2)^{3/2}}=−\frac{x}{a^2\sqrt{x^2−a^2}}+C$$

### Integrals Involving a2 − x2, a > 0

85. $$\quad \displaystyle ∫\sqrt{a^2-x^2}\,dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a}+C$$

86. $$\quad \displaystyle ∫x^2\sqrt{a^2-x^2}\,dx=\frac{x}{8}(2x^2−a^2)\sqrt{a^2-x^2}+\frac{a^4}{8}\arcsin\frac{x}{a}+C$$

87. $$\quad \displaystyle ∫\frac{\sqrt{a^2-x^2}}{x}\,dx=\sqrt{a^2-x^2}−a\ln \left|\frac{a+\sqrt{a^2-x^2}}{x}\right|+C$$

88. $$\quad \displaystyle ∫\frac{\sqrt{a^2-x^2}}{x^2}\,dx=\frac{−1}{x}\sqrt{a^2-x^2}−\arcsin\frac{x}{a}+C$$

89. $$\quad \displaystyle ∫\frac{x^2}{\sqrt{a^2-x^2}}\,dx=\frac{1}{2}\left(-x\sqrt{a^2-x^2}+a^2\arcsin \frac{x}{a}\right)+C$$

90. $$\quad \displaystyle ∫\frac{dx}{x\sqrt{a^2-x^2}}=−\frac{1}{a}\ln \left|\frac{a+\sqrt{a^2-x^2}}{x}\right|+C$$

91. $$\quad \displaystyle ∫\frac{dx}{x^2\sqrt{a^2-x^2}}=−\frac{1}{a^2x}\sqrt{a^2-x^2}+C$$

92. $$\quad \displaystyle ∫\left(a^2−x^2\right)^{3/2}\,dx=−\frac{x}{8}\left(2x^2−5a^2\right)\sqrt{a^2-x^2}+\frac{3a^4}{8}\arcsin \frac{x}{a}+C$$

93. $$\quad \displaystyle ∫\frac{dx}{(a^2−x^2)^{3/2}}=−\frac{x}{a^2\sqrt{a^2−x^2}}+C$$

### Integrals Involving a2 − x2, x2 < a2

94. $$\quad \displaystyle ∫\frac{dx}{(a^2−x^2)}=\frac{1}{a}\tanh^{-1}{\frac{x}{a}}$$

95. $$\quad \displaystyle ∫\frac{x^3 dx}{(a^2−x^2)^2}=\frac{x}{2(a^2-x^2)} - \frac{1}{2}\ln(a^2-x^2)$$

96. $$\quad \displaystyle ∫\frac{x dx}{(a^2−x^2)^n}=\frac{1}{2(n-1)(a^2-x^2)^{n-1}}$$

### Integrals Involving 2ax − x2, a > 0

97. $$\quad \displaystyle ∫\sqrt{2ax−x^2}\,dx=\frac{x−a}{2}\sqrt{2ax−x^2}+\frac{a^2}{2}\arccos\left(\frac{a−x}{a}\right)+C$$

98. $$\quad \displaystyle ∫\frac{dx}{\sqrt{2ax−x^2}}=\arccos\left(\frac{a−x}{a}\right)+C$$

99. $$\quad \displaystyle ∫x\sqrt{2ax−x^2}\,dx=\frac{2x^2−ax−3a^2}{6}\sqrt{2ax−x^2}+\frac{a^3}{2}\arccos\left(\frac{a−x}{a}\right)+C$$

100. $$\quad \displaystyle ∫\frac{dx}{x\sqrt{2ax−x^2}}=−\frac{\sqrt{2ax−x^2}}{ax}+C$$

### Integrals Involving a + bx, a ≠ 0

101. $$\quad \displaystyle ∫\frac{x}{a+bx}\,dx=\frac{1}{b^2}(a+bx−a\ln |a+bx|)+C$$

102. $$\quad \displaystyle ∫\frac{x^2}{a+bx}\,dx=\frac{1}{2b^3}\left[(a+bx)^2−4a(a+bx)+2a^2\ln |a+bx|\right]+C$$

103. $$\quad \displaystyle ∫\frac{dx}{x(a+bx)}=\frac{1}{a}\ln \left|\frac{x}{a+bx}\right|+C$$

104. $$\quad \displaystyle ∫\frac{dx}{x^2(a+bx)}=−\frac{1}{ax}+\frac{b}{a^2}\ln \left|\frac{a+bx}{x}\right|+C$$

105. $$\quad \displaystyle ∫\frac{x}{(a+bx)^2}\,dx=\frac{a}{b^2(a+bx)}+\frac{1}{b^2}\ln |a+bx|+C$$

106. $$\quad \displaystyle ∫\frac{x}{x(a+bx)^2}\,dx=\frac{1}{a(a+bx)}−\frac{1}{a^2}\ln \left|\frac{a+bx}{x}\right|+C$$

107. $$\quad \displaystyle ∫\frac{x^2}{(a+bx)^2}\,dx=\frac{1}{b^3}\left(a+bx−\frac{a^2}{a+bx}−2a\ln |a+bx|\right)+C$$

108. $$\quad \displaystyle ∫x\sqrt{a+bx}\,dx=\frac{2}{15b^2}(3bx−2a)(a+bx)^{3/2}+C$$

109. $$\quad \displaystyle ∫\frac{x}{\sqrt{a+bx}}\,dx=\frac{2}{3b^2}(bx−2a)\sqrt{a+bx}+C$$

110. $$\quad \displaystyle ∫\frac{x^2}{\sqrt{a+bx}}\,dx=\frac{2}{15b^3}(8a^2+3b^2x^2−4abx)\sqrt{a+bx}+C$$

111. $$\quad \displaystyle ∫\frac{dx}{x\sqrt{a+bx}}=\begin{cases} \frac{1}{\sqrt{a}}\ln \left|\frac{\sqrt{a+bx}−\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right|+C,\quad \text{if}\,a>0\\[5pt] \frac{\sqrt{2}}{\sqrt{−a}}\arctan\sqrt{\frac{a+bx}{−a}}+C,\quad \text{if}\,a<0 \end{cases}$$

112. $$\quad \displaystyle ∫\frac{\sqrt{a+bx}}{x}\,dx=2\sqrt{a+bx}+a∫\frac{dx}{x\sqrt{a+bx}}$$

113. $$\quad \displaystyle ∫\frac{\sqrt{a+bx}}{x^2}\,dx=−\frac{\sqrt{a+bx}}{x}+\frac{b}{2}∫\frac{dx}{x\sqrt{a+bx}}$$

114. $$\quad \displaystyle ∫x^n\sqrt{a+bx}\,dx=\frac{2}{b(2n+3)}\left[x^n(a+bx)^{3/2}−na∫x^{n−1}\sqrt{a+bx}\,dx\right]$$

115. $$\quad \displaystyle ∫\frac{x^n}{\sqrt{a+bx}}\,dx=\frac{2x^n\sqrt{a+bx}}{b(2n+1)}−\frac{2na}{b(2n+1)}∫\frac{x^{n−1}}{\sqrt{a+bx}}\,dx$$

116. $$\quad \displaystyle ∫\frac{dx}{x^n\sqrt{a+bx}}=−\frac{\sqrt{a+bx}}{a(n−1)x^{n−1}}−\frac{b(2n−3)}{2a(n−1)}∫\frac{dx}{x^{n-1}\sqrt{a+bx}}$$

For other integrals you could use the computational intelligence website Wolframalpha.