Popis integrala eksponencijalnih funkcija

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

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

$\int a^{cx}dx=\frac{1}{c\ln a}{a}^{cx}+C\text{(za }a>0,a\ne 1\text{)}$

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

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

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

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

$\int \frac{e^{cx}}{x^n}dx=\frac{1}{n-1}(-\frac{e^{cx}}{x^{n-1}}+c\int \frac{e^{cx}}{x^{n-1}}dx)+C\text{(za }n\ne 1\text{)}$

$\int e^{cx}\ln xdx=\frac{1}{c}{e}^{cx}\ln |x|-\mathrm{Ei}(cx)+C$

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

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

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

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

$\int x{e}^{c{x}^2}dx=\frac{1}{2c}{e}^{c{x}^2}+C$

$\int e^{-c{x}^2}dx=\sqrt{\frac{\pi }{4c}}\mathrm{erf}(\sqrt{c}x)+C$($\text{erf}$ je funkcija grješke (error function))

$\int x{e}^{-c{x}^2}dx=-\frac{1}{2c}{e}^{-c{x}^2}+C$

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

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

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

Određeni integrali

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-a{x}^2}dx=\frac{1}{2}\sqrt{\frac{\pi }{a}}(a>0)$ (Gaussov integral)

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-a{x}^2}dx=\sqrt{\frac{\pi }{a}}(a>0)$

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-a{x}^2}{e}^{2bx}dx=\sqrt{\frac{\pi }{a}}{e}^{\frac{b^2}{a}}(a>0)$

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x{e}^{-a(x-b{)}^2}dx=b\sqrt{\frac{\pi }{a}}(a>0)$

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^2{e}^{-a{x}^2}dx=\frac{1}{2}\sqrt{\frac{\pi }{a^3}}(a>0)$

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

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^n{e}^{-ax}dx=\{\begin{array}{cc}\frac{\mathrm{\Gamma }(n+1)}{a^{n+1}}& (n>-1,a>0)\\ \frac{n!}{a^{n+1}}& (n=0,1,2,\dots ,a>0)\end{array}$

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-ax}\sin bxdx=\frac{b}{a^2+b^2}(a>0)$

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-ax}\cos bxdx=\frac{a}{a^2+b^2}(a>0)$

$\underset{0}{\overset{\mathrm{\infty }}{\int }}x{e}^{-ax}\sin bxdx=\frac{2ab}{(a^2+b^2{)}^2}(a>0)$

$\underset{0}{\overset{\mathrm{\infty }}{\int }}x{e}^{-ax}\cos bxdx=\frac{a^2-b^2}{(a^2+b^2{)}^2}(a>0)$

$\underset{0}{\overset{2\pi }{\int }}{e}^{x\cos \theta }d\theta =2\pi I_0(x)$ ($I_0$ je modificirana Besselova funkcija prve vrste)

$\underset{0}{\overset{2\pi }{\int }}{e}^{x\cos \theta +y\sin \theta }d\theta =2\pi I_0\left(\sqrt{x^2+y^2}\right)$