Seznam integrálů exponenciálních funkcí

Toto je seznam integrálů (primitivních funkcí) exponenciálních funkcí.

${\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}e^{cx}$

${\displaystyle \int a^{cx}\;\mathrm {d} x={\frac {1}{c\ln a}a^{cx}\qquad {\mbox{(pro }a>0,{\mbox{ }a\neq 1{\mbox{)}$

${\displaystyle \int xe^{cx}\;\mathrm {d} x={\frac {e^{cx}{c^{2}(cx-1)}$

${\displaystyle \int x^{2}e^{cx}\;\mathrm {d} x=e^{cx}\left({\frac {x^{2}{c}-{\frac {2x}{c^{2}+{\frac {2}{c^{3}\right)}$

${\displaystyle \int x^{n}e^{cx}\;\mathrm {d} x={\frac {1}{c}x^{n}e^{cx}-{\frac {n}{c}\int x^{n-1}e^{cx}\mathrm {d} x}$

${\displaystyle \int {\frac {e^{cx}\;\mathrm {d} x}{x}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}{i\cdot i!}$

${\displaystyle \int {\frac {e^{cx}\;\mathrm {d} x}{x^{n}={\frac {1}{n-1}\left(-{\frac {e^{cx}{x^{n-1}+c\int {\frac {e^{cx}{x^{n-1}\,\mathrm {d} x\right)\qquad {\mbox{(pro }n\neq 1{\mbox{)}$

${\displaystyle \int e^{cx}\ln x\;\mathrm {d} x={\frac {1}{c}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)}$

${\displaystyle \int e^{cx}\sin bx\;\mathrm {d} x={\frac {e^{cx}{c^{2}+b^{2}(c\sin bx-b\cos bx)}$

${\displaystyle \int e^{cx}\cos bx\;\mathrm {d} x={\frac {e^{cx}{c^{2}+b^{2}(c\cos bx+b\sin bx)}$

${\displaystyle \int e^{cx}\sin ^{n}x\;\mathrm {d} x={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}\int e^{cx}\sin ^{n-2}x\;\mathrm {d} x}$

${\displaystyle \int e^{cx}\cos ^{n}x\;\mathrm {d} x={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}\int e^{cx}\cos ^{n-2}x\;\mathrm {d} x}$

${\displaystyle \int xe^{cx^{2}\;\mathrm {d} x={\frac {1}{2c}\;e^{cx^{2}$

${\displaystyle \int {1 \over \sigma {\sqrt {2\pi }\,e^{-{(x-\mu )^{2}/2\sigma ^{2}\;dx={\frac {1}{2\sigma }(1+{\mbox{erf}\,{\frac {x-\mu }{\sigma {\sqrt {2})}$

${\displaystyle \int e^{x^{2}\,\mathrm {d} x=e^{x^{2}\left(\sum _{j=0}^{n-1}c_{2j}\,{\frac {1}{x^{2j+1}\right)+(2n-1)c_{2n-2}\int {\frac {e^{x^{2}{x^{2n}\;\mathrm {d} x\quad {\mbox{platí pro }n>0,}$

kde ${\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}={\frac {2j\,!}{j!\,2^{2j+1}\ .}$

${\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}\,\mathrm {d} x={\sqrt {\pi \over a}$ (Gaussův integrál)

${\displaystyle \int _{0}^{\infty }x^{2n}e^{-{x^{2}/{a^{2}\,\mathrm {d} x={\sqrt {\pi }{(2n)! \over {n!}{\left({\frac {a}{2}\right)}^{2n+1}$

${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }\mathrm {d} \theta =2\pi I_{0}(x)}$ (${\displaystyle I_{0}$ je modifikovaná Besselova funkce prvního druhu)

${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }\mathrm {d} \theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}\right)}$