本稿は指数関数を含む式の原始関数の一覧である。
不定積分
以下は不定積分の一覧である。右辺につく積分定数は省略している。
![{\displaystyle \int e^{x}\;\mathrm {d} x=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09bfd7870d2bc343fa190eedff345df5c1e9607d)
![{\displaystyle \int f'(x)e^{f(x)}\;\mathrm {d} x=e^{f(x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b073e6bf355a53b24469c5545a26371669a1e025)
![{\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8a2d8d0ed5abc7ccb0bd31adf4274be7dd54ea)
(
)
![{\displaystyle \int xe^{cx}\;\mathrm {d} x={\frac {e^{cx}}{c^{2}}}(cx-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/500198d16fb049f067c35b7e7cb88d132fcb7783)
![{\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/5e267454873adf7adc1827062f44cad4d17488c2)
![{\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=\left({\frac {\partial }{\partial c}}\right)^{n}{\frac {e^{cx}}{c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd11f7411115b696c2c79b971f6f7773b3cbce05)
![{\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/5c73c96bbd063b40cd6bb1ced9c96f8d98a5de8f)
![{\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{(}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/598ffec96cdaf8b471aa5474b002843c5daa6b7d)
![{\displaystyle \int e^{cx}\ln x\;\mathrm {d} x={\frac {1}{c}}\left(e^{cx}\ln |x|-\operatorname {Ei} \,(cx)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09efcfeec0db1b4feed49557386f8b622d7a0c76)
![{\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/95923c67929de601326519b6d3691da8753eb74f)
![{\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/1df7aee04b3bc5dade2f76b500551f19e43ee7b9)
![{\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/d8f62ae0a81bf35f4e80680484429d46032ac888)
![{\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/a91207df8692dd303615aee89b17de3c86b72079)
![{\displaystyle \int xe^{cx^{2}}\;\mathrm {d} x={\frac {1}{2c}}\;e^{cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaad8f6cee041e8e3ce3a611f8ebaf9df994ce1b)
(
は誤差関数)
![{\displaystyle \int xe^{-cx^{2}}\;\mathrm {d} x=-{\frac {1}{2c}}e^{-cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30467d4ac29c992b8fd62db75447783e08e374df)
![{\displaystyle \int {\frac {e^{-x^{2}}}{x^{2}}}\;\mathrm {d} x=-{\frac {e^{-x^{2}}}{x}}-{\sqrt {\pi }}\mathrm {erf} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/111d80f6c931225a9e5b339dfb012fba6c24fca5)
![{\displaystyle \int {{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}\;\mathrm {d} x={\frac {1}{2}}\left(\operatorname {erf} \,{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d617e3f0726b343188ff777820169425d7c0e20)
- ここで
とする。
- ここで
![{\displaystyle a_{mn}={\begin{cases}1&n=0,\\{\frac {1}{n!}}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{otherwise}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7882e11ad4424c8ba10681bcd9cbe03ee90b9ed)
はガンマ関数、
はクヌースの矢印表記、
はコンウェイのチェーン表記
(
,
, かつ
)
(
,
, かつ
)
定積分
(
)
![{\displaystyle \int _{0}^{\infty }e^{ax}\,\mathrm {d} x={\frac {1}{-a}}\quad (\operatorname {Re} (a)<0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5aad5beee968042e51edb9eb2715ee08c9c0939)
(ガウス積分)
![{\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)
(ガウス関数の積分)
![{\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,\mathrm {d} x=b{\sqrt {\frac {\pi }{a}}}\quad (\operatorname {Re} (a)>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72d0bfdbd7c4375e0b446eff356b0021dafafec8)
![{\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)
(!! は二重階乗)
![{\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^{b}}dx={\frac {1}{b}}\ a^{-{\frac {1}{b}}}\,\Gamma \left({\frac {1}{b}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ec6200b217ea3c5dcab17e1f1c10d81e06153a)
![{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {n+1}{b}}}\,\Gamma \left({\frac {n+1}{b}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecca8baa65309a7a6396807e9d842a43c8d1c96e)
![{\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)
(
は変形ベッセル関数)
![{\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)
脚注
- Wolfram Mathematica Online Integrator
- V. H. Moll, The Integrals in Gradshteyn and Ryzhik