Popis integrala eksponencijalnih funkcija

Slijedi popis integrala (antiderivacija funkcija) eksponencijalnih funkcija. Za potpun popis integrala funkcija, pogledati tablica integrala i popis integrala.

${\displaystyle \int e^{cx}\;dx={\frac {1}{c}e^{cx}+C}$, ali ${\displaystyle \int e^{2x}\;dx=2e^{2x}+C}$

${\displaystyle \int a^{cx}\;dx={\frac {1}{c\ln a}a^{cx}+C\qquad {\mbox{(za }a>0,{\mbox{ }a\neq 1{\mbox{)}$

${\displaystyle \int xe^{cx}\;dx={\frac {e^{cx}{c^{2}(cx-1)+C}$

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

${\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}x^{n}e^{cx}-{\frac {n}{c}\int x^{n-1}e^{cx}dx+C}$

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

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

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

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

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

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

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

${\displaystyle \int xe^{cx^{2}\;dx={\frac {1}{2c}\;e^{cx^{2}+C}$

${\displaystyle \int e^{-cx^{2}\;dx={\sqrt {\frac {\pi }{4c}\operatorname {erf} ({\sqrt {c}x)+C}$(${\displaystyle {\mbox{erf}$ je funkcija grješke (error function))

${\displaystyle \int xe^{-cx^{2}\;dx=-{\frac {1}{2c}e^{-cx^{2}+C}$

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

${\displaystyle \int e^{x^{2}\,dx=e^{x^{2}\left(\sum _{j=0}^{n-1}c_{2j}\,{\frac {1}{x^{2j+1}\right)+(2n-1)c_{2n-2}\int {\frac {e^{x^{2}{x^{2n}\;dx+C\quad {\mbox{valjano za }n>0,}$

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

${\displaystyle \int _{0}^{\infty }e^{-ax^{2}\,dx={\frac {1}{2}{\sqrt {\pi \over a}\quad (a>0)}$ (Gaussov integral)

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

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

${\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}\,dx=b{\sqrt {\pi \over a}\quad (a>0)}$

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

${\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{2}\,dx={\begin{cases}{\frac {1}{2}\Gamma \left({\frac {n+1}{2}\right)/a^{\frac {n+1}{2}&(n>-1,a>0)\\{\frac {(2k-1)!!}{2^{k+1}a^{k}{\sqrt {\frac {\pi }{a}&(n=2k,k\;{\text{cijeli broj},a>0)\\{\frac {k!}{2a^{k+1}&(n=2k+1,k\;{\text{cijeli broj},a>0)\end{cases}$ (!! je dvostruka faktorijela)

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

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

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

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

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

${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)}$ (${\displaystyle I_{0}$ je modificirana Besselova funkcija prve vrste)

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