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In mathematics, the definite integral
![{\displaystyle \int _{a}^{b}f(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac02adeed584466d53dee65f3228ad66939eb58b)
is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals.
If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures. for example:
![{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\left[\int _{a}^{b}f(x)\,dx\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a095fbd3f49faf63702fb3b314abd0e76c40f8a)
A constant, such pi, that may be defined by the integral of an algebraic function over an algebraic domain is known as a period.
The following is a list of some of the most common or interesting definite integrals. For a list of indefinite integrals see List of indefinite integrals.
Definite integrals involving rational or irrational expressions
![{\displaystyle \int _{0}^{\infty }{\frac {dx}{1+x^{p}}}={\frac {\pi /p}{\sin(\pi /p)}}\quad {\text{for }}\Re (p)>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1736791d464d4e6ab8ddd44421ed5413129405b3)
![{\displaystyle \int _{0}^{\infty }{\frac {x^{p-1}dx}{1+x}}={\frac {\pi }{\sin(p\pi )}}\quad {\text{for }}0<p<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17bd64f1a86eb6f4af2b612eb195e1c378869464)
![{\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{x^{n}+a^{n}}}={\frac {\pi a^{m-n+1}}{n\sin \left({\dfrac {m+1}{n}}\pi \right)}}\quad {\text{for }}0<m+1<n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/480ffc020d11ef5650324dd961b99ebb50132c1b)
![{\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{1+2x\cos \beta +x^{2}}}={\frac {\pi }{\sin(m\pi )}}\cdot {\frac {\sin(m\beta )}{\sin(\beta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2176c1aa0829382ca27a0b211edc5e0a10d7319)
![{\displaystyle \int _{0}^{a}{\frac {dx}{\sqrt {a^{2}-x^{2}}}}={\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1818d048a5d3cb5aa101f5012255eed3a1377914)
![{\displaystyle \int _{0}^{a}{\sqrt {a^{2}-x^{2}}}dx={\frac {\pi a^{2}}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0de64c418875ae187444692b04df41dbf4e08ceb)
![{\displaystyle \int _{0}^{a}x^{m}(a^{n}-x^{n})^{p}\,dx={\frac {a^{m+1+np}\Gamma \left({\dfrac {m+1}{n}}\right)\Gamma (p+1)}{n\Gamma \left({\dfrac {m+1}{n}}+p+1\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/663945bf263305fd27efd7a16735fa12070a0fdd)
![{\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{({x^{n}+a^{n})}^{r}}}={\frac {(-1)^{r-1}\pi a^{m+1-nr}\Gamma \left({\dfrac {m+1}{n}}\right)}{n\sin \left({\dfrac {m+1}{n}}\pi \right)(r-1)!\,\Gamma \left({\dfrac {m+1}{n}}-r+1\right)}}\quad {\text{for }}n(r-2)<m+1<nr}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4121403d8607319048eed21238ff44ea282e1515)
Definite integrals involving trigonometric functions
![{\displaystyle \int _{0}^{\pi }\sin(mx)\sin(nx)dx={\begin{cases}0&{\text{if }}m\neq n\\\\{\dfrac {\pi }{2}}&{\text{if }}m=n\end{cases}}\quad {\text{for }}m,n{\text{ positive integers}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9c3bff999e6aeec56581034e934689c14e8206)
![{\displaystyle \int _{0}^{\pi }\cos(mx)\cos(nx)dx={\begin{cases}0&{\text{if }}m\neq n\\\\{\dfrac {\pi }{2}}&{\text{if }}m=n\end{cases}}\quad {\text{for }}m,n{\text{ positive integers}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff2c6a64e691dc111e16b3a544bc41b6d36a907a)
![{\displaystyle \int _{0}^{\pi }\sin(mx)\cos(nx)dx={\begin{cases}0&{\text{if }}m+n{\text{ even}}\\\\{\dfrac {2m}{m^{2}-n^{2}}}&{\text{if }}m+n{\text{ odd}}\end{cases}}\quad {\text{for }}m,n{\text{ integers}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22e1cc455b1d7a70e7bcddbad945508ac3db01b4)
![{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2}(x)dx=\int _{0}^{\frac {\pi }{2}}\cos ^{2}(x)dx={\frac {\pi }{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1978f1ecf7c2b39966340043ed6e18c244ad39)
![{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2m}(x)dx=\int _{0}^{\frac {\pi }{2}}\cos ^{2m}(x)dx={\frac {1\times 3\times 5\times \cdots \times (2m-1)}{2\times 4\times 6\times \cdots \times 2m}}\cdot {\frac {\pi }{2}}\quad {\text{for }}m=1,2,3\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8100c09a1cdfb4a37d48491489710c6c77a5f208)
![{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}(x)dx=\int _{0}^{\frac {\pi }{2}}\cos ^{2m+1}(x)dx={\frac {2\times 4\times 6\times \cdots \times 2m}{1\times 3\times 5\times \cdots \times (2m+1)}}\quad {\text{for }}m=1,2,3\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/991d3165419b4ce71062925be02036d196ed1407)
![{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2p-1}(x)\cos ^{2q-1}(x)dx={\frac {\Gamma (p)\Gamma (q)}{2\Gamma (p+q)}}={\frac {1}{2}}{\text{B}}(p,q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2074289d402519c0f9ffb9e9423128ab0ee4cc85)
(see Dirichlet integral) ![{\displaystyle \int _{0}^{\infty }{\frac {\sin px\cos qx}{x}}\ dx={\begin{cases}0&{\text{ if }}q>p>0\\\\{\dfrac {\pi }{2}}&{\text{ if }}0<q<p\\\\{\dfrac {\pi }{4}}&{\text{ if }}p=q>0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e5c07506906a7c6330530b2201748fa3145c54)
![{\displaystyle \int _{0}^{\infty }{\frac {\sin px\sin qx}{x^{2}}}\ dx={\begin{cases}{\dfrac {\pi p}{2}}&{\text{ if }}0<p\leq q\\\\{\dfrac {\pi q}{2}}&{\text{ if }}0<q\leq p\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f67aa8c1a0cb161aee8b0c5662e1c3a3fd99b9)
![{\displaystyle \int _{0}^{\infty }{\frac {\sin ^{2}px}{x^{2}}}\ dx={\frac {\pi p}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a273d93a80e0c9d89b27507531a604e997f8952e)
![{\displaystyle \int _{0}^{\infty }{\frac {1-\cos px}{x^{2}}}\ dx={\frac {\pi p}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eff7c5341f53379b86527eacf85d14bc292fb81f)
![{\displaystyle \int _{0}^{\infty }{\frac {\cos px-\cos qx}{x}}\ dx=\ln {\frac {q}{p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69a1b31e8afa7e21d57f6175c87d5fd6d8118208)
![{\displaystyle \int _{0}^{\infty }{\frac {\cos px-\cos qx}{x^{2}}}\ dx={\frac {\pi (q-p)}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/256a77dec12001d35e37438aabc2535179795cbd)
![{\displaystyle \int _{0}^{\infty }{\frac {\cos mx}{x^{2}+a^{2}}}\ dx={\frac {\pi }{2a}}e^{-ma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b64615cf5336a9ec83d4fdb05aa7c5553393d9)
![{\displaystyle \int _{0}^{\infty }{\frac {x\sin mx}{x^{2}+a^{2}}}\ dx={\frac {\pi }{2}}e^{-ma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85e71e45154a52de384c585d740deb6cfffef4f8)
![{\displaystyle \int _{0}^{\infty }{\frac {\sin mx}{x(x^{2}+a^{2})}}\ dx={\frac {\pi }{2a^{2}}}\left(1-e^{-ma}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2eb6d1d6291ab54df2ceb5e2d444c35f10343b8)
![{\displaystyle \int _{0}^{2\pi }{\frac {dx}{a+b\sin x}}={\frac {2\pi }{\sqrt {a^{2}-b^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ab1c90537c535f6180979a6852ccb716c1f670)
![{\displaystyle \int _{0}^{2\pi }{\frac {dx}{a+b\cos x}}={\frac {2\pi }{\sqrt {a^{2}-b^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f915121c00aaf995cd4419b56a3897527c9579d0)
![{\displaystyle \int _{0}^{\frac {\pi }{2}}{\frac {dx}{a+b\cos x}}={\frac {\cos ^{-1}\left({\dfrac {b}{a}}\right)}{\sqrt {a^{2}-b^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24af5072b4b8667249cc8f15ae7d3a3f1c47b982)
![{\displaystyle \int _{0}^{2\pi }{\frac {dx}{(a+b\sin x)^{2}}}=\int _{0}^{2\pi }{\frac {dx}{(a+b\cos x)^{2}}}={\frac {2\pi a}{(a^{2}-b^{2})^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a15fa89d460b76ab0f70e0a40843c02fb0d07e)
![{\displaystyle \int _{0}^{2\pi }{\frac {dx}{1-2a\cos x+a^{2}}}={\frac {2\pi }{1-a^{2}}}\quad {\text{for }}0<a<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38e52090dd892ee1b3f974e7e8f76634465a66a6)
![{\displaystyle \int _{0}^{\pi }{\frac {x\sin x\ dx}{1-2a\cos x+a^{2}}}={\begin{cases}{\dfrac {\pi }{a}}\ln \left|1+a\right|&{\text{if }}|a|<1\\\\{\dfrac {\pi }{a}}\ln \left|1+{\dfrac {1}{a}}\right|&{\text{if }}|a|>1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14b97dbd9427a38f41aaf803ad22a60799bf279b)
![{\displaystyle \int _{0}^{\pi }{\frac {\cos mx\ dx}{1-2a\cos x+a^{2}}}={\frac {\pi a^{m}}{1-a^{2}}}\quad {\text{for }}a^{2}<1\ ,\ m=0,1,2,\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d26e59ac34a0d69afd1eeb5015ebd17c36d77419)
![{\displaystyle \int _{0}^{\infty }\sin ax^{2}\ dx=\int _{0}^{\infty }\cos ax^{2}={\frac {1}{2}}{\sqrt {\frac {\pi }{2a}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d431dc06209153f7df9ccc0e5fd30d5bcdca9df5)
![{\displaystyle \int _{0}^{\infty }\sin ax^{n}={\frac {1}{na^{1/n}}}\Gamma \left({\frac {1}{n}}\right)\sin {\frac {\pi }{2n}}\quad {\text{for }}n>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2cde4f446b6e2dbe0f4b22a4d3e850265e8eb5)
![{\displaystyle \int _{0}^{\infty }\cos ax^{n}={\frac {1}{na^{1/n}}}\Gamma \left({\frac {1}{n}}\right)\cos {\frac {\pi }{2n}}\quad {\text{for }}n>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/729333e4615ce3b5123305f347434197caad0116)
![{\displaystyle \int _{0}^{\infty }{\frac {\sin x}{\sqrt {x}}}\ dx=\int _{0}^{\infty }{\frac {\cos x}{\sqrt {x}}}\ dx={\sqrt {\frac {\pi }{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a206ae00ece0f3c38a8aec70a74580ac5fce56)
![{\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x^{p}}}\ dx={\frac {\pi }{2\Gamma (p)\sin \left({\dfrac {p\pi }{2}}\right)}}\quad {\text{for }}0<p<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2cb8f72675b3483ab69fc81bed11f639183416f)
![{\displaystyle \int _{0}^{\infty }{\frac {\cos x}{x^{p}}}\ dx={\frac {\pi }{2\Gamma (p)\cos \left({\dfrac {p\pi }{2}}\right)}}\quad {\text{for }}0<p<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b213ef5fed2a4b06e2de192bd864f0cbee0ee1)
![{\displaystyle \int _{0}^{\infty }\sin ax^{2}\cos 2bx\ dx={\frac {1}{2}}{\sqrt {\frac {\pi }{2a}}}\left(\cos {\frac {b^{2}}{a}}-\sin {\frac {b^{2}}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9458363d2144a34e48c4654c65e0628f6d396990)
![{\displaystyle \int _{0}^{\infty }\cos ax^{2}\cos 2bx\ dx={\frac {1}{2}}{\sqrt {\frac {\pi }{2a}}}\left(\cos {\frac {b^{2}}{a}}+\sin {\frac {b^{2}}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc32763b8ffa898414d8e65d1950087eeef15a99)
Definite integrals involving exponential functions
(see also Gamma function) ![{\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx\,dx={\frac {a}{a^{2}+b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9397a3fbfd8834a91947b440871943d996fc6a54)
![{\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx\,dx={\frac {b}{a^{2}+b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c818588bc61e1b3ffcf81966f9dc0012d04f15d2)
![{\displaystyle \int _{0}^{\infty }{\frac {{}e^{-ax}\sin bx}{x}}\,dx=\tan ^{-1}{\frac {b}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf0e229ce8f0bb3f73a5de677c992a569ef69eb)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}}{x}}\,dx=\ln {\frac {b}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2175f6ab38d3bcbda8248ff8539181b3374e4aed)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-\cos(bx)}{x}}\,dx=\ln {\frac {b}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cda3900dcc4ffbe3446af9a0cc1c4b904969966d)
(the Gaussian integral) ![{\displaystyle \int _{0}^{\infty }{e^{-ax^{2}}}\cos bx\,dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a}}}e^{\left({\frac {-b^{2}}{4a}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87b3f03c2e3a051a7ff2aabf3c82ba5c8a2d1477)
![{\displaystyle \int _{0}^{\infty }e^{-(ax^{2}+bx+c)}\,dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a}}}e^{\left({\frac {b^{2}-4ac}{4a}}\right)}\cdot \operatorname {erfc} {\frac {b}{2{\sqrt {a}}}},{\text{ where }}\operatorname {erfc} (p)={\frac {2}{\sqrt {\pi }}}\int _{p}^{\infty }e^{-x^{2}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d084facc2501c46c4e95326109c8ff86a0b9d23d)
![{\displaystyle \int _{-\infty }^{\infty }e^{-(ax^{2}+bx+c)}\ dx={\sqrt {\frac {\pi }{a}}}e^{\left({\frac {b^{2}-4ac}{4a}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b12e5db89da962c0c51c4e6926942fc418fbc446)
![{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax}\ dx={\frac {\Gamma (n+1)}{a^{n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfee4665fa6e144d84df517b754aead0792b1c1e)
![{\displaystyle \int _{0}^{\infty }{x^{2}e^{-ax^{2}}\,dx}={\frac {1}{4}}{\sqrt {\frac {\pi }{a^{3}}}}\quad {\text{for }}a>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5724e6febebe05824b425b74537609b3a362cd1e)
(where !! is the double factorial) ![{\displaystyle \int _{0}^{\infty }{x^{3}e^{-ax^{2}}\,dx}={\frac {1}{2a^{2}}}\quad {\text{for }}a>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dea44bb4d070093ac17707559e46ccf13abe1d9)
![{\displaystyle \int _{0}^{\infty }x^{2n+1}e^{-ax^{2}}\,dx={\frac {n}{a}}\int _{0}^{\infty }x^{2n-1}e^{-ax^{2}}\,dx={\frac {n!}{2a^{n+1}}}\quad {\text{for }}a>0\ ,\ n=0,1,2\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/01f5ae43a599676330b5e71c218d9a77574a78fc)
![{\displaystyle \int _{0}^{\infty }x^{m}e^{-ax^{2}}\ dx={\frac {\Gamma \left({\dfrac {m+1}{2}}\right)}{2a^{\left({\frac {m+1}{2}}\right)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c8fa912cfd274ce246e31e9e611cf5a9d649bd)
![{\displaystyle \int _{0}^{\infty }e^{\left(-ax^{2}-{\frac {b}{x^{2}}}\right)}\ dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a}}}e^{-2{\sqrt {ab}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26a69c2d0c99dc9a7abd27e49186c9b182abfb82)
![{\displaystyle \int _{0}^{\infty }{\frac {x}{e^{x}-1}}\ dx=\zeta (2)={\frac {\pi ^{2}}{6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2300c15b15af5b9c9472d386987e1ed3592b74e)
![{\displaystyle \int _{0}^{\infty }{\frac {x^{n-1}}{e^{x}-1}}\ dx=\Gamma (n)\zeta (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce515a3618ee547c6bbc75a89e75dc7da5a69a9f)
![{\displaystyle \int _{0}^{\infty }{\frac {x}{e^{x}+1}}\ dx={\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}-{\frac {1}{4^{2}}}+\dots ={\frac {\pi ^{2}}{12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea6a872215107ba71acbcdb7b09286f5b4226d49)
![{\displaystyle \int _{0}^{\infty }{\frac {x^{n}}{e^{x}+1}}\ dx=n!\cdot \left({\frac {2^{n}-1}{2^{n}}}\right)\zeta (n+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f71ba2e4cacb84652fb8b9052bcbf8550fab7b31)
![{\displaystyle \int _{0}^{\infty }{\frac {\sin mx}{e^{2\pi x}-1}}\ dx={\frac {1}{4}}\coth {\frac {m}{2}}-{\frac {1}{2m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d55e7fa36074690dece3cefef9f05c79932cc21e)
(where
is Euler–Mascheroni constant) ![{\displaystyle \int _{0}^{\infty }{\frac {e^{-x^{2}}-e^{-x}}{x}}\ dx={\frac {\gamma }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15d183d4e5aa745e0596b348e7a4834d1020de40)
![{\displaystyle \int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {e^{-x}}{x}}\right)\ dx=\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/21ccecdc557673f30839ded08e2bb0aa0bf20944)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}}{x\sec px}}\ dx={\frac {1}{2}}\ln {\frac {b^{2}+p^{2}}{a^{2}+p^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bbf2c445775e8e02fcc8f8f2fa2519a955ad321)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}}{x\csc px}}\ dx=\tan ^{-1}{\frac {b}{p}}-\tan ^{-1}{\frac {a}{p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7c4174249ed42784c1d2109aedc3d2ea29677c)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}(1-\cos x)}{x^{2}}}\ dx=\cot ^{-1}a-{\frac {a}{2}}\ln \left|{\frac {a^{2}+1}{a^{2}}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc45c5f1790fd62fce521c5eea530e9fd6fd939)
![{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44592478ca6a48493154e57210fcedd304814e3a)
![{\displaystyle \int _{-\infty }^{\infty }x^{2(n+1)}e^{-{\frac {1}{2}}x^{2}}\,dx={\frac {(2n+1)!}{2^{n}n!}}{\sqrt {2\pi }}\quad {\text{for }}n=0,1,2,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1a137d287ea0b79eb8390fa2bfb781b8c65891)
Definite integrals involving logarithmic functions
![{\displaystyle \int _{0}^{1}x^{m}(\ln x)^{n}\,dx={\frac {(-1)^{n}n!}{(m+1)^{n+1}}}\quad {\text{for }}m>-1,n=0,1,2,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/14dce7deeeb2df4a1dded6c1d5f92b092b68a339)
![{\displaystyle \int _{1}^{\infty }x^{m}(\ln x)^{n}\,dx={\frac {(-1)^{n+1}n!}{(m+1)^{n+1}}}\quad {\text{for }}m<-1,n=0,1,2,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fa22a7567033a92debafde7f43a33e038cb04f9)
![{\displaystyle \int _{0}^{1}{\frac {\ln x}{1+x}}\,dx=-{\frac {\pi ^{2}}{12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6250bd808b27f1cb32f675d5edce096a4f800a28)
![{\displaystyle \int _{0}^{1}{\frac {\ln x}{1-x}}\,dx=-{\frac {\pi ^{2}}{6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e04a08e132ca2c26bd213a2d75b4721a76b15eba)
![{\displaystyle \int _{0}^{1}{\frac {\ln(1+x)}{x}}\,dx={\frac {\pi ^{2}}{12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aff6d22f78246d9d0541a5097fdabebb9d86b468)
![{\displaystyle \int _{0}^{1}{\frac {\ln(1-x)}{x}}\,dx=-{\frac {\pi ^{2}}{6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca8168283ea8f2fbeaf7c894ed6aa7a878e6c4d)
![{\displaystyle \int _{0}^{\infty }{\frac {\ln(a^{2}+x^{2})}{b^{2}+x^{2}}}\ dx={\frac {\pi }{b}}\ln(a+b)\quad {\text{for }}a,b>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9293cb25e9cefd578f0f12393d2e26e7c3f53a0f)
![{\displaystyle \int _{0}^{\infty }{\frac {\ln x}{x^{2}+a^{2}}}\ dx={\frac {\pi \ln a}{2a}}\quad {\text{for }}a>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3cf8a02f1df1d62411226e57f37d8bb0385160)
Definite integrals involving hyperbolic functions
![{\displaystyle \int _{0}^{\infty }{\frac {\sin ax}{\sinh bx}}\ dx={\frac {\pi }{2b}}\tanh {\frac {a\pi }{2b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5710b835195ace9f5e9728bf18eeb963539af36)
![{\displaystyle \int _{0}^{\infty }{\frac {\cos ax}{\cosh bx}}\ dx={\frac {\pi }{2b}}\cdot {\frac {1}{\cosh {\frac {a\pi }{2b}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/493254abcabda9ab41e18d5ee516a42025d451a0)
![{\displaystyle \int _{0}^{\infty }{\frac {x}{\sinh ax}}\ dx={\frac {\pi ^{2}}{4a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49ee7a4c18d1e929adfb0f3c1717641a14856afa)
![{\displaystyle \int _{0}^{\infty }{\frac {x^{2n+1}}{\sinh ax}}\ dx=c_{2n+1}\left({\frac {\pi }{a}}\right)^{2(n+1)},\quad c_{2n+1}={\frac {(-1)^{n}}{2}}\left({\frac {1}{2}}-\sum _{k=0}^{n-1}(-1)^{k}{2n+1 \choose 2k+1}c_{2k+1}\right),\quad c_{1}={\frac {1}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cff733b50ad3dbc9fe2b4df1738b54126239fdb1)
![{\displaystyle \int _{0}^{\infty }{\frac {1}{\cosh ax}}\ dx={\frac {\pi }{2a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a867485f04f8dc7fddc885142536ac8154b649b)
![{\displaystyle \int _{0}^{\infty }{\frac {x^{2n}}{\cosh ax}}\ dx=d_{2n}\left({\frac {\pi }{a}}\right)^{2n+1},\quad d_{2n}={\frac {(-1)^{n}}{2}}\left({\frac {1}{4^{n}}}-\sum _{k=0}^{n-1}(-1)^{k}{2n \choose 2k}d_{2k}\right),\quad d_{0}={\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/207b0c30a09c7e08efacc009ab0ad35157320c6e)
![{\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\ dx=\left(\lim _{x\to 0}f(x)-\lim _{x\to \infty }f(x)\right)\ln \left({\frac {b}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f2fb9d161e0e35295cf567cc3df9cb01778918)
holds if the integral exists and
![{\displaystyle f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd7d7c75340e779d82658e19d1720ce84ab127)
is continuous.
See also
Mathematics portal
References
- Reynolds, Robert; Stauffer, Allan (2020). "Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions". Mathematics. 8 (687): 687. doi:10.3390/math8050687.
- Reynolds, Robert; Stauffer, Allan (2019). "A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function". Mathematics. 7 (1148): 1148. doi:10.3390/math7121148.
- Reynolds, Robert; Stauffer, Allan (2019). "Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series". Mathematics. 7 (1099): 1099. doi:10.3390/math7111099.
- Winckler, Anton (1861). "Eigenschaften Einiger Bestimmten Integrale". Hof, K.K., Ed.
- Spiegel, Murray R.; Lipschutz, Seymour; Liu, John (2009). Mathematical handbook of formulas and tables (3rd ed.). McGraw-Hill. ISBN 978-0071548557.
- Zwillinger, Daniel (2003). CRC standard mathematical tables and formulae (32nd ed.). CRC Press. ISBN 978-143983548-7.
- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.