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Mathematics LibreTexts

Table of Integrals

  • Page ID
    14736
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    Basic Integrals

    1. \(\quad \displaystyle ∫u^n\,du=\frac{u^{n+1}}{n+1}+C,\quad n≠−1\)

    2. \(\quad \displaystyle ∫\frac{du}{u} =\ln |u|+C\)

    3. \(\quad \displaystyle ∫e^u\,du=e^u+C\)

    4. \(\quad \displaystyle ∫a^u\,du=\frac{a^u}{\ln a}+C\)

    5. \(\quad \displaystyle ∫\sin u\,du=−\cos u+C\)

    6. \(\quad \displaystyle ∫\cos u\,du=\sin u+C\)

    7. \(\quad \displaystyle ∫\sec^2u\,du=\tan u+C\)

    8. \(\quad \displaystyle ∫\csc^2u\,du=−\cot u+C\)

    9. \(\quad \displaystyle ∫\sec u\tan u\,du=\sec u+C\)

    10. \(\quad \displaystyle ∫\csc u\cot u\,du=−\csc u+C\)

    11. \(\quad \displaystyle ∫\tan u\,du=\ln |\sec u|+C\)

    12. \(\quad \displaystyle ∫\cot u\,du=\ln |\sin u|+C\)

    13. \(\quad \displaystyle ∫\sec u\,du=\ln |\sec u+\tan u|+C\)

    14. \(\quad \displaystyle ∫\csc u\,du=\ln |\csc u−\cot u|+C\)

    15. \(\quad \displaystyle ∫\frac{du}{\sqrt{a^2−u^2}}=\sin^{−1}\left(\frac{u}{a}\right)+C\)

    16. \(\quad \displaystyle ∫\frac{du}{a^2+u^2}=\frac{1}{a}\tan^{−1}\left(\frac{u}{a}\right)+C\)

    17. \(\quad \displaystyle ∫\frac{du}{u\sqrt{u^2−a^2}}=\frac{1}{a}\sec^{−1}\frac{|u|}{a}+C\)

    Trigonometric Integrals

    18. \(\quad \displaystyle ∫\sin^2u\,du=\frac{1}{2}u−\frac{1}{4}\sin 2u+C\)

    19. \(\quad \displaystyle ∫\cos^2 u\,du=\frac{1}{2}u+\frac{1}{4}\sin 2u+C\)

    20. \(\quad \displaystyle ∫\tan^2 u\,du=\tan u−u+C\)

    21. \(\quad \displaystyle ∫\cot ^2 u\,du=−\cot u−u+C\)

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

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

    24. \(\quad \displaystyle ∫\tan^3 u\,du=\frac{1}{2}\tan^2 u+\ln |\cos u|+C\)

    25. \(\quad \displaystyle ∫\cot^3 u\,du=−\frac{1}{2}\cot^2 u−\ln |\sin u|+C\)

    26. \(\quad \displaystyle ∫\sec^3 u\,du=\frac{1}{2}\sec u\tan u+\frac{1}{2}\ln |\sec u+\tan u|+C\)

    27. \(\quad \displaystyle ∫\csc^3 u\,du=−\frac{1}{2}\csc u\cot u+\frac{1}{2}\ln |\csc u−\cot u|+C\)

    28. \(\quad \displaystyle ∫\sin^n u\,du=\frac{-1}{n}\sin^{n−1}u\cos u+\frac{n−1}{n}∫\sin^{n−2}u\,du\)

    29. \(\quad \displaystyle ∫\cos^n u\,du=\frac{1}{n}\cos^{n−1} u\sin u+\frac{n−1}{n}∫\cos^{n−2}u\,du\)

    30. \(\quad \displaystyle ∫\tan^n u\,du=\frac{1}{n-1}\tan^{n−1} u−∫\tan^{n−2} u\,du\)

    31. \(\quad \displaystyle ∫\cot^n u\,du=\frac{-1}{n-1}\cot^{n−1}u−∫\cot^{n−2}u\,du\)

    32. \(\quad \displaystyle ∫\sec^n u\,du=\frac{1}{n-1}\tan u\sec^{n−2}u+\frac{n-2}{n-1}∫\sec^{n−2}u\,du\)

    33. \(\quad \displaystyle ∫\csc^n u\,du=\frac{-1}{n-1}\cot u\csc^{n−2}u+\frac{n-2}{n-1}∫\csc^{n−2}u\,du\)

    34. \(\quad \displaystyle ∫\sin au\sin bu\,du=\frac{\sin (a−b)u}{2(a−b)}−\frac{\sin (a+b)u}{2(a+b)}+C\)

    35. \(\quad \displaystyle ∫\cos au\cos bu\,du=\frac{\sin (a−b)u}{2(a−b)}+\frac{\sin (a+b)u}{2(a+b)}+C\)

    36. \(\quad \displaystyle ∫\sin au\cos bu\,du=−\frac{\cos (a−b)u}{2(a−b)}−\frac{\cos (a+b)u}{2(a+b)}+C\)

    37. \(\quad \displaystyle ∫u\sin u\,du=\sin u−u\cos u+C\)

    38. \(\quad \displaystyle ∫u\cos u\,du=\cos u+u\sin u+C\)

    39. \(\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du\)

    40. \(\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du\)

    41. \(\quad \begin{align*} \displaystyle ∫\sin^n u\cos^m u\,du &= −\frac{\sin^{n−1}u\cos^{m+1}u}{n+m}+\frac{n−1}{n+m}∫\sin^{n−2}u\cos^m u\,du \\[5pt] &=\frac{\sin^{n+1}u\cos^{m−1}u}{n+m}+\frac{m−1}{n+m}∫\sin^n u\cos^{m−2}u \,du \end{align*}\)

    Exponential and Logarithmic Integrals

    42. \(\quad \displaystyle ∫ue^{au}\,du=\frac{1}{a^2}(au−1)e^{au}+C\)

    43. \(\quad \displaystyle ∫u^ne^{au}\,du=\frac{1}{a}u^ne^{au}−\frac{n}{a}∫u^{n−1}e^{au}\,du\)

    44. \(\quad \displaystyle ∫e^{au}\sin bu\,du=\frac{e^{au}}{a^2+b^2}(a\sin bu−b\cos bu)+C\)

    45. \(\quad \displaystyle ∫e^{au}\cos bu\,du=\frac{e^{au}}{a^2+b^2}(a\cos bu+b\sin bu)+C\)

    46. \(\quad \displaystyle ∫\ln u\,du=u\ln u−u+C\)

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

    48. \(\quad \displaystyle ∫\frac{1}{u\ln u}\,du=\ln |\ln u|+C\)

    Hyperbolic Integrals

    49. \(\quad \displaystyle ∫\sinh u\,du=\cosh u+C\)

    50. \(\quad \displaystyle ∫\cosh u\,du=\sinh u+C\)

    51. \(\quad \displaystyle ∫\tanh u\,du=\ln \cosh u+C\)

    52. \(\quad \displaystyle ∫\coth u\,du=\ln |\sinh u|+C\)

    53. \(\quad \displaystyle ∫\text{sech}\,u\,du=\tan^{−1}|\sinh u|+C\)

    54. \(\quad \displaystyle ∫\text{csch}\,u\,du=\ln ∣\tanh\frac{1}{2}u∣+C\)

    55. \(\quad \displaystyle ∫\text{sech}^2 u\,du=\tanh \,u+C\)

    56. \(\quad \displaystyle ∫\text{csch}^2 u\,du=−\coth \,u+C\)

    57. \(\quad \displaystyle ∫\text{sech} \,u\tanh u\,du=−\text{sech} \,u+C\)

    58. \(\quad \displaystyle ∫\text{csch} \,u\coth u\,du=−\text{csch} \,u+C\)

    Inverse Trigonometric Integrals

    59. \(\quad \displaystyle ∫\sin^{-1}u\,du=u\sin^{-1}u+\sqrt{1−u^2}+C\)

    60. \(\quad \displaystyle ∫\cos^{-1}u\,du=u\cos^{-1}u−\sqrt{1−u^2}+C\)

    61. \(\quad \displaystyle ∫\tan^{-1}u\,du=u\tan^{-1}u−\frac{1}{2}\ln (1+u^2)+C\)

    62. \(\quad \displaystyle ∫u\sin^{-1}u\,du=\frac{2u^2−1}{4}\sin^{-1}u+\frac{u\sqrt{1−u^2}}{4}+C\)

    63. \(\quad \displaystyle ∫u\cos^{-1}u\,du=\frac{2u^2−1}{4}\cos^{-1}u-\frac{u\sqrt{1−u^2}}{4}+C\)

    64. \(\quad \displaystyle ∫u\tan^{-1}u\,du=\frac{u^2+1}{2}\tan^{-1}u−\frac{u}{2}+C\)

    65. \(\quad \displaystyle ∫u^n\sin^{-1}u\,du=\frac{1}{n+1}\left[u^{n+1}\sin^{-1}u−∫\frac{u^{n+1}\,du}{\sqrt{1−u^2}}\right],\quad n≠−1\)

    66. \(\quad \displaystyle ∫u^n\cos^{-1}u\,du=\frac{1}{n+1}\left[u^{n+1}\cos^{-1}u+∫\frac{u^{n+1}\,du}{\sqrt{1−u^2}}\right],\quad n≠−1\)

    67. \(\quad \displaystyle ∫u^n\tan^{-1}u\,du=\frac{1}{n+1}\left[u^{n+1}\tan^{-1}u−∫\frac{u^{n+1}\,du}{1+u^2}\right],\quad n≠−1\)

    Integrals Involving a2 + u2a > 0

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

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

    70. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u}\,du=\sqrt{a^2+u^2}−a\ln \left|\frac{a+\sqrt{a^2+u^2}}{u}\right|+C\)

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

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

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

    74. \(\quad \displaystyle ∫\frac{du}{u\sqrt{a^2+u^2}}=\frac{−1}{a}\ln \left|\frac{\sqrt{a^2+u^2}+a}{u}\right|+C\)

    75. \(\quad \displaystyle ∫\frac{du}{u^2\sqrt{a^2+u^2}}=−\frac{\sqrt{a^2+u^2}}{a^2u}+C\)

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

    Integrals Involving u2 − a2a > 0

    77. \(\quad \displaystyle ∫\sqrt{u^2−a^2}\,du=\frac{u}{2}\sqrt{u^2−a^2}−\frac{a^2}{2}\ln \left|u+\sqrt{u^2−a^2}\right|+C\)

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

    79. \(\quad \displaystyle ∫\frac{\sqrt{u^2−a^2}}{u}\,du=\sqrt{u^2−a^2}−a\cos^{-1}\frac{a}{|u|}+C\)

    80. \(\quad \displaystyle ∫\frac{\sqrt{u^2−a^2}}{u^2}\,du=−\frac{\sqrt{u^2−a^2}}{u}+\ln \left|u+\sqrt{u^2−a^2}\right|+C\)

    81. \(\quad \displaystyle ∫\frac{du}{\sqrt{u^2−a^2}}=\ln \left|u+\sqrt{u^2−a^2}\right|+C\)

    82. \(\quad \displaystyle ∫\frac{u^2}{\sqrt{u^2−a^2}}\,du=\frac{u}{2}\sqrt{u^2−a^2}+\frac{a^2}{2}\ln \left|u+\sqrt{u^2−a^2}\right|+C\)

    83. \(\quad \displaystyle ∫\frac{du}{u^2\sqrt{u^2−a^2}}=\frac{\sqrt{u^2−a^2}}{a^2u}+C\)

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

    Integrals Involving a2 − u2a > 0

    85. \(\quad \displaystyle ∫\sqrt{a^2-u^2}\,du=\frac{u}{2}\sqrt{a^2-u^2}+\frac{a^2}{2}\sin^{-1}\frac{u}{a}+C\)

    86. \(\quad \displaystyle ∫u^2\sqrt{a^2-u^2}\,du=\frac{u}{8}(2u^2−a^2)\sqrt{a^2-u^2}+\frac{a^4}{8}\sin^{-1}\frac{u}{a}+C\)

    87. \(\quad \displaystyle ∫\frac{\sqrt{a^2-u^2}}{u}\,du=\sqrt{a^2-u^2}−a\ln \left|\frac{a+\sqrt{a^2-u^2}}{u}\right|+C\)

    88. \(\quad \displaystyle ∫\frac{\sqrt{a^2-u^2}}{u^2}\,du=\frac{−1}{u}\sqrt{a^2-u^2}−\sin^{-1}\frac{u}{a}+C\)

    89. \(\quad \displaystyle ∫\frac{u^2}{\sqrt{a^2-u^2}}\,du=\frac{1}{2}\left(-u\sqrt{a^2-u^2}+a^2\sin^{-1}\frac{u}{a}\right)+C\)

    90. \(\quad \displaystyle ∫\frac{du}{u\sqrt{a^2-u^2}}=−\frac{1}{a}\ln \left|\frac{a+\sqrt{a^2-u^2}}{u}\right|+C\)

    91. \(\quad \displaystyle ∫\frac{du}{u^2\sqrt{a^2-u^2}}=−\frac{1}{a^2u}\sqrt{a^2-u^2}+C\)

    92. \(\quad \displaystyle ∫\left(a^2−u^2\right)^{3/2}\,du=−\frac{u}{8}\left(2u^2−5a^2\right)\sqrt{a^2-u^2}+\frac{3a^4}{8}\sin^{-1}\frac{u}{a}+C\)

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

    Integrals Involving 2au − u2a > 0

    94. \(\quad \displaystyle ∫\sqrt{2au−u^2}\,du=\frac{u−a}{2}\sqrt{2au−u^2}+\frac{a^2}{2}\cos^{-1}\left(\frac{a−u}{a}\right)+C\)

    95. \(\quad \displaystyle ∫\frac{du}{\sqrt{2au−u^2}}=\cos^{-1}\left(\frac{a−u}{a}\right)+C\)

    96. \(\quad \displaystyle ∫u\sqrt{2au−u^2}\,du=\frac{2u^2−au−3a^2}{6}\sqrt{2au−u^2}+\frac{a^3}{2}\cos^{-1}\left(\frac{a−u}{a}\right)+C\)

    97. \(\quad \displaystyle ∫\frac{du}{u\sqrt{2au−u^2}}=−\frac{\sqrt{2au−u^2}}{au}+C\)

    Integrals Involving a + bua ≠ 0

    98. \(\quad \displaystyle ∫\frac{u}{a+bu}\,du=\frac{1}{b^2}(a+bu−a\ln |a+bu|)+C\)

    99. \(\quad \displaystyle ∫\frac{u^2}{a+bu}\,du=\frac{1}{2b^3}\left[(a+bu)^2−4a(a+bu)+2a^2\ln |a+bu|\right]+C\)

    100. \(\quad \displaystyle ∫\frac{du}{u(a+bu)}=\frac{1}{a}\ln \left|\frac{u}{a+bu}\right|+C\)

    101. \(\quad \displaystyle ∫\frac{du}{u^2(a+bu)}=−\frac{1}{au}+\frac{b}{a^2}\ln \left|\frac{a+bu}{u}\right|+C\)

    102. \(\quad \displaystyle ∫\frac{u}{(a+bu)^2}\,du=\frac{a}{b^2(a+bu)}+\frac{1}{b^2}\ln |a+bu|+C\)

    103. \(\quad \displaystyle ∫\frac{u}{u(a+bu)^2}\,du=\frac{1}{a(a+bu)}−\frac{1}{a^2}\ln \left|\frac{a+bu}{u}\right|+C\)

    104. \(\quad \displaystyle ∫\frac{u^2}{(a+bu)^2}\,du=\frac{1}{b^3}\left(a+bu−\frac{a^2}{a+bu}−2a\ln |a+bu|\right)+C\)

    105. \(\quad \displaystyle ∫u\sqrt{a+bu}\,du=\frac{2}{15b^2}(3bu−2a)(a+bu)^{3/2}+C\)

    106. \(\quad \displaystyle ∫\frac{u}{\sqrt{a+bu}}\,du=\frac{2}{3b^2}(bu−2a)\sqrt{a+bu}+C\)

    107. \(\quad \displaystyle ∫\frac{u^2}{\sqrt{a+bu}}\,du=\frac{2}{15b^3}(8a^2+3b^2u^2−4abu)\sqrt{a+bu}+C\)

    108. \(\quad \displaystyle ∫\frac{du}{u\sqrt{a+bu}}=\begin{cases} \frac{1}{\sqrt{a}}\ln \left|\frac{\sqrt{a+bu}−\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}\right|+C,\quad \text{if}\,a>0\\[5pt] \frac{\sqrt{2}}{\sqrt{−a}}\tan^{-1}\sqrt{\frac{a+bu}{−a}}+C,\quad \text{if}\,a<0 \end{cases}\)

    109. \(\quad \displaystyle ∫\frac{\sqrt{a+bu}}{u}\,du=2\sqrt{a+bu}+a∫\frac{du}{u\sqrt{a+bu}}\)

    110. \(\quad \displaystyle ∫\frac{\sqrt{a+bu}}{u^2}\,du=−\frac{\sqrt{a+bu}}{u}+\frac{b}{2}∫\frac{du}{u\sqrt{a+bu}}\)

    111. \(\quad \displaystyle ∫u^n\sqrt{a+bu}\,du=\frac{2}{b(2n+3)}\left[u^n(a+bu)^{3/2}−na∫u^{n−1}\sqrt{a+bu}\,du\right]\)

    112. \(\quad \displaystyle ∫\frac{u^n}{\sqrt{a+bu}}\,du=\frac{2u^n\sqrt{a+bu}}{b(2n+1)}−\frac{2na}{b(2n+1)}∫\frac{u^{n−1}}{\sqrt{a+bu}}\,du\)

    113. \(\quad \displaystyle ∫\frac{du}{u^n\sqrt{a+bu}}=−\frac{\sqrt{a+bu}}{a(n−1)u^{n−1}}−\frac{b(2n−3)}{2a(n−1)}∫\frac{du}{u^{n-1}\sqrt{a+bu}}\)

     

    Contributors

    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.