Anexo:Integrales de funciones exponenciales

La siguiente es una lista de integrales de funciones exponenciales (Agréguese a cada integral una constante arbitraria).

$\int a^{cx}dx=\frac{1}{c\ln a}{a}^{cx}$

$\int e^{cx}dx=\frac{1}{c}{e}^{cx}$

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

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

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

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

$\int e^{{ax}^2\mathrm{+}bx\mathrm{+}c}dx\mathrm{=}\mathrm{-}\frac{i}{e^{\frac{b\mathrm{-}4ac}{4a}}}\sqrt{a}\frac{\mathit{\pi }}{2}erf\left(\frac{2ax\mathrm{+}b}{2\sqrt{a}}i\right)con\mathrm{△}>0ya>0$

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

$\int e^{cx}\ln xdx=\frac{1}{c}(e^{cx}\ln |x|-\int \frac{e^{cx}dx}{x})$

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

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

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

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

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

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

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^{2n}{e}^{-\alpha x^2}dx=(-1{)}^n\frac{d^n}{d{\alpha }^n}I(\alpha )\stackrel{\text{si n = 1}}{\to }=\frac{\sqrt{\pi }}{2{\alpha }^{3/2}}$