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14.8.1: Table of Integrals

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    60838
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    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 x2a2, 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.

    Contributors and Attributions

    • Template:ContribOpenStaxCalc
    • Inverse trig notation changes by Paxl Seebxrger (Monroe Community College)
    • Change u to x as the u for students was confusing by Scott Johnson (PGCC)

    14.8.1: Table of Integrals is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.