Lista de integrais de funções exponenciais

Predefinição:Sem-fontes A lista seguinte contém integrais de funções exponenciais.

Integral Exponencial:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\int e^{x^2}dx=e^{x^2}\left(\sum _{r=1}^n\frac{1}{2^n{x}^{2n-1}}\right)+\frac{2n-1}{2^n}\int \frac{e^{x^2}dx}{x^{2n}}$

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