Lijst van integralen van exponentiële functies

Dit artikel bevat een lijst van integralen van exponentiële functies.

Onbepaalde integralen

In de onderstaande betrekkingen is $c$ een willekeurige reëel getal.

$\int e^{x}\,\mathrm {d} x=e^{x}$

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

$\int a^{cx}\,\mathrm {d} x={\frac {1}{c\cdot \ln a}}a^{cx}$ voor $a>0,\ a\neq 1$

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

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

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

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

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

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

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

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

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

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

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

$\int {\sqrt {e^{cx}}}\,\mathrm {d} x={\frac {2{\sqrt {e^{cx}}}}{c}}$

$\int {\sqrt {e^{cx^{n}}}}\,\mathrm {d} x={\frac {{\sqrt[{n}]{2}}xe^{-{\frac {cx^{n}}{2}}}{\sqrt {e^{cx^{n}}}}\Gamma \left({\frac {1}{n}},-{\frac {cx^{n}}{2}}\right)}{n{\sqrt[{n}]{-cx^{n}}}}}$

$\int e^{-cx^{2}}\,\mathrm {d} x={\sqrt {\frac {\pi }{4c}}}\mathrm {erf} ({\sqrt {c}}x)\quad$ ($\mathrm {erf}$ is de zogenaamde errorfunctie)

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

$\int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\,\mathrm {d} x={\frac {1}{2}}\left(1+\mathrm {erf} \,{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)$

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

waarbij $c_{2j}={\frac {1\cdot 3\cdot 5\ldots (2j-1)}{2^{j+1}}}={\frac {(2j)\,!}{j!\,2^{2j+1}}}$

${\int \underbrace {x^{x^{\cdot ^{\cdot ^{x}}}}} _{m}\,\mathrm {d} x=\sum _{n=0}^{m}{\frac {(-1)^{n}(n+1)^{n-1}}{n!}}\Gamma (n+1,-\ln x)+\sum _{n=m+1}^{\infty }(-1)^{n}a_{mn}\Gamma (n+1,-\ln x)\qquad {\mbox{voor }}x>0}$

waarbij $a_{mn}={\begin{cases}1&{\text{als }}n=0,\\{\frac {1}{n!}}&{\text{als }}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{alle andere gevallen}}\end{cases}}$

Bepaalde integralen

$\int _{0}^{1}e^{x\cdot \ln a+(1-x)\cdot \ln b}\;\mathrm {d} x=\int _{0}^{1}\left({\frac {a}{b}}\right)^{x}\cdot b\;\mathrm {d} x=\int _{0}^{1}a^{x}\cdot b^{1-x}\;\mathrm {d} x={\frac {a-b}{\ln a-\ln b}}$ voor $a>0,\ b>0,\ a\neq b$

$\int _{0}^{\infty }e^{-ax}\,\mathrm {d} x={\frac {1}{a}}$

$\int _{0}^{\infty }e^{-ax^{2}}\,\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a}}\quad (a>0)$ (de Gauss-integraal)

$\int _{-\infty }^{\infty }e^{-ax^{2}}\,\mathrm {d} x={\sqrt {\pi \over a}}\quad (a>0)$

$\int _{-\infty }^{\infty }e^{-ax^{2}}e^{-2bx}\,\mathrm {d} x={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{a}}\quad (a>0)$

$\int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,\mathrm {d} x=b{\sqrt {\frac {\pi }{a}}}$

$\int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}\quad (a>0)$

$\int _{0}^{\infty }x^{n}e^{-ax^{2}}\,\mathrm {d} x={\begin{cases}{\frac {1}{2}}\Gamma \left({\frac {n+1}{2}}\right)/a^{\frac {n+1}{2}}&(n>-1,a>0)\\{\frac {(2k-1)!!}{2^{k+1}a^{k}}}{\sqrt {\frac {\pi }{a}}}&(n=2k,{\mbox{met }}k\in \mathbb {Z} ,a>0)\\{\frac {k!}{2a^{k+1}}}&(n=2k+1,{\mbox{met }}k\in \mathbb {Z} ,a>0)\end{cases}}$ (!! is de dubbelfaculteit)

$\int _{0}^{\infty }x^{n}e^{-ax}\,\mathrm {d} x={\begin{cases}{\frac {\Gamma (n+1)}{a^{n+1}}}&(n>-1,a>0)\\{\frac {n!}{a^{n+1}}}&(n=0,1,2,\ldots ,a>0)\\\end{cases}}$

$\int _{0}^{\infty }e^{-ax}\sin bx\,\mathrm {d} x={\frac {b}{a^{2}+b^{2}}}\quad (a>0)$

$\int _{0}^{\infty }e^{-ax}\cos bx\,\mathrm {d} x={\frac {a}{a^{2}+b^{2}}}\quad (a>0)$

$\int _{0}^{\infty }xe^{-ax}\sin bx\,\mathrm {d} x={\frac {2ab}{(a^{2}+b^{2})^{2}}}\quad (a>0)$

$\int _{0}^{\infty }xe^{-ax}\cos bx\,\mathrm {d} x={\frac {a^{2}-b^{2}}{(a^{2}+b^{2})^{2}}}\quad (a>0)$

$\int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)$ ($I_{0}$ is de Besselfunctie van de eerste graad)

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