Uit Wikipedia, de vrije encyclopedie
Dit artikel bevat een lijst van integralen van exponentiële functies.
Onbepaalde integralen
In de onderstaande betrekkingen is
een willekeurige reëel getal.
![{\displaystyle \int e^{x}\,\mathrm {d} x=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6144fc6257a2ce689b9564f498642874ee8f63)
![{\displaystyle \int e^{cx}\,\mathrm {d} x={\frac {1}{c}}e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ecb589d8b41060d153288c757ac617e462c0304)
voor ![{\displaystyle a>0,\ a\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c27d65d4a22ff9cb56ae8562f0ba1ee2b4a7f120)
![{\displaystyle \int xe^{cx}\,\mathrm {d} x={\frac {e^{cx}}{c^{2}}}(cx-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8186d0be2ed64c58540baa8a0692f93df344c4)
![{\displaystyle \int x^{2}e^{cx}\,\mathrm {d} x=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5ac326bd6e7cc97a7ccbac8917619e468f3b1a)
![{\displaystyle \int x^{n}e^{cx}\,\mathrm {d} x={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}\,\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61ef0b4517ee5f67c0249d733edeb437257114be)
![{\displaystyle \int {\frac {e^{cx}}{x}}\,\mathrm {d} x=\ln |x|+\sum _{n=1}^{\infty }{\frac {(cx)^{n}}{n\cdot n!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e7c5d610e34a90eec7146dcf2a761153b7f5033)
![{\displaystyle \int {\frac {e^{cx}}{x^{n}}}\,\mathrm {d} x={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,\mathrm {d} x\right)\qquad {\mbox{voor }}n\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c803a1309d12b06197e911eb6fad6f8c25950bf)
![{\displaystyle \int e^{cx}\ln x\,\mathrm {d} x={\frac {1}{c}}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff45566d8ea06d9897aeb1553fea6d6eecfb880)
![{\displaystyle \int e^{cx}\sin bx\,\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92a94b66d51954461af09ab006240495010cdfb6)
![{\displaystyle \int e^{cx}\cos bx\,\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84aedac926d45037fcae81cecf21bd7394f28ad2)
![{\displaystyle \int e^{cx}\sin ^{n}x\,\mathrm {d} x={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\,\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdebaec5900b0c2435bfc080a92bb419232585da)
![{\displaystyle \int e^{cx}\cos ^{n}x\,\mathrm {d} x={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\,\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61fc7c13e1ef35432e1d2de016b0c69178a4fc86)
![{\displaystyle \int xe^{cx^{2}}\,\mathrm {d} x={\frac {1}{2c}}\,e^{cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd87b232d7a462a522d45318f532939fb06bb697)
![{\displaystyle \int {\sqrt {e^{cx}}}\,\mathrm {d} x={\frac {2{\sqrt {e^{cx}}}}{c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0682cbd6f1f7448cb29ffa37630346d84f4afe3)
![{\displaystyle \int {\sqrt {e^{cx^{n}}}}\,\mathrm {d} x={\frac {{\sqrt[{n}]{2}}xe^{-{\frac {cx^{n}}{2}}}{\sqrt {e^{cx^{n}}}}\Gamma \left({\frac {1}{n}},-{\frac {cx^{n}}{2}}\right)}{n{\sqrt[{n}]{-cx^{n}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4737a0507f2ab6ffb8f99c743edd7f17aff84bf)
(
is de zogenaamde errorfunctie)
![{\displaystyle \int xe^{-cx^{2}}\,\mathrm {d} x=-{\frac {1}{2c}}e^{-cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2218e58389758609195eded3afdebcc14e6ade)
![{\displaystyle \int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\,\mathrm {d} x={\frac {1}{2}}\left(1+\mathrm {erf} \,{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c49adf226d5a806492db246db482ad3114ec5d5)
,
- waarbij
![{\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\ldots (2j-1)}{2^{j+1}}}={\frac {(2j)\,!}{j!\,2^{2j+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72dfe0b0d73665579d933080d063c1ae310bed95)
![{\displaystyle {\int \underbrace {x^{x^{\cdot ^{\cdot ^{x}}}}} _{m}\,\mathrm {d} x=\sum _{n=0}^{m}{\frac {(-1)^{n}(n+1)^{n-1}}{n!}}\Gamma (n+1,-\ln x)+\sum _{n=m+1}^{\infty }(-1)^{n}a_{mn}\Gamma (n+1,-\ln x)\qquad {\mbox{voor }}x>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8045c0f3d1f337d2e36637d4deda4ba53d3326e)
- waarbij
![{\displaystyle a_{mn}={\begin{cases}1&{\text{als }}n=0,\\{\frac {1}{n!}}&{\text{als }}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{alle andere gevallen}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c084a4f7702b685bb9c4366a0dcf119e15089b)
Bepaalde integralen
voor ![{\displaystyle a>0,\ b>0,\ a\neq b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6922fdd26fd761e8f5856f5eed9b52fb9a0eb38)
![{\displaystyle \int _{0}^{\infty }e^{-ax}\,\mathrm {d} x={\frac {1}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7ed3a8e6dd52dc11d0a83e31105a0244bf2e68)
(de Gauss-integraal)
![{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}\,\mathrm {d} x={\sqrt {\pi \over a}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/726b7008c42f0da6d3d9b4b3df8bcb2aac6b4ac6)
![{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}e^{-2bx}\,\mathrm {d} x={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{a}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c20f129d3d7143bd6d0e646ceb520292645107e)
![{\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,\mathrm {d} x=b{\sqrt {\frac {\pi }{a}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aae114874193bba246042bdadc40cb01053f038b)
![{\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/034c82717067aa92e4672ae9ca69d5e317e12f22)
(!! is de dubbelfaculteit)
![{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax}\,\mathrm {d} x={\begin{cases}{\frac {\Gamma (n+1)}{a^{n+1}}}&(n>-1,a>0)\\{\frac {n!}{a^{n+1}}}&(n=0,1,2,\ldots ,a>0)\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e56ff6ccc4fcb5b354ad33de0cfda66500454c6)
![{\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx\,\mathrm {d} x={\frac {b}{a^{2}+b^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/116b9485eb78a3df65cac57937643c8091ddfb7e)
![{\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx\,\mathrm {d} x={\frac {a}{a^{2}+b^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3823f932a06e10b125bfe342f2baa1023cdaa350)
![{\displaystyle \int _{0}^{\infty }xe^{-ax}\sin bx\,\mathrm {d} x={\frac {2ab}{(a^{2}+b^{2})^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e3b468a59eb2d4b6daa62060b7e3f52ad3da2f)
![{\displaystyle \int _{0}^{\infty }xe^{-ax}\cos bx\,\mathrm {d} x={\frac {a^{2}-b^{2}}{(a^{2}+b^{2})^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55434eae15e366659f77b386ed7056af99478a)
(
is de Besselfunctie van de eerste graad)
![{\displaystyle \int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc7da0077239149468cbcc5eb3576109c8d0d4d)