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Mathematics LibreTexts

Table of Integrals

  • Page ID
    14727
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    For this course, all work must be shown to obtain most of these integral forms. Of the integration formulas listed below, the only ones that can be applied without further work are #1 - 10, 15 - 17, and 49 and 50. And even these will require work to be shown if a substitution is involved.

    All others may be helpful for checking your final answers, but cannot be used to skip the necessary work to show you understand how to use the integration techniques taught in this course.

    As you look through these formulas, you should be able to recognize which integration technique was needed to obtain the general formula. It may be very useful for you to try to obtain the general formula yourself using the techniques we learn in this course.

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

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

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

    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. \(\quad \displaystyle ∫\sin^n u\cos^m u\,du=\) Use the methods for powers of sine and cosine

    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. \(\quad \displaystyle ∫\text{sech}\,u\,du=\arctan |\sinh u|+C\)

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

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

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

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

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

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

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

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

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

    Integrals Involving a2 + u2, a > 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 u2a2, a > 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. \(\quad \displaystyle ∫\frac{\sqrt{u^2−a^2}}{u}\,du=\sqrt{u^2−a^2}−a\arccos\frac{a}{|u|}+C\)

    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 a2u2, a > 0

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

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

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

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

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

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

    Integrals Involving 2auu2, a > 0

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

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

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

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

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

    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.

    • Introduction paragraphs and inverse trig notation changes by Paul Seeburger (Monroe Community College)