From Wikibooks, open books for an open world
Jump to navigation
Jump to search
![{\displaystyle \int e^{x}\;\mathrm {d} x=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09bfd7870d2bc343fa190eedff345df5c1e9607d)
![{\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8a2d8d0ed5abc7ccb0bd31adf4274be7dd54ea)
for ![{\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/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{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3edd292ca8743cc942cd94d98458d3b720a085ce)
![{\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/ac3ea804792382d0e90e6af19b854052e67e0530)
![{\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)
(
is the Error function)
![{\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 {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;\mathrm {d} x=-{\frac {1}{2}}\left({\mbox{erf}}\,{\frac {-x+\mu }{\sigma {\sqrt {2}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01c65288310b896343923850fb1cb172067bd953)
- where
![{\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {(2j)\,!}{j!\,2^{2j+1}}}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90dd5e5695126e16ae330b936f2ae1abc90fcf3a)
- where
![{\displaystyle a_{mn}={\begin{cases}1&{\text{if }}n=0,\\{\frac {1}{n!}}&{\text{if }}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/45d61d68551812ed0abe96cad49132369d763677)
- and
is the Gamma Function
when
,
, and ![{\displaystyle ae^{\lambda x}+b>0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3728b822a9eafe02b95482cf00c7182f01eb43)
when
,
, and ![{\displaystyle ae^{\lambda x}+b>0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3728b822a9eafe02b95482cf00c7182f01eb43)
for
, which is the logarithmic mean
![{\displaystyle \int _{0}^{\infty }e^{ax}\,\mathrm {d} x={\frac {1}{a}}(a<0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85d16a6f9c642231142ef93736b2ac9933cf1966)
(the Gaussian integral)
![{\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)
(see Integral of a Gaussian function)
![{\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 the double factorial)
![{\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 the modified Bessel function of the first kind)
![{\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)