Integral de $$$x e^{- x^{2}}$$$

Encontre $$$\int x e^{- x^{2}}\, dx$$$.

Seja $$$u=- x^{2}$$$.

Então $$$du=\left(- x^{2}\right)^{\prime }dx = - 2 x dx$$$ (veja os passos »), e obtemos $$$x dx = - \frac{du}{2}$$$.

Portanto,

$${\color{red}{\int{x e^{- x^{2}} d x}}} = {\color{red}{\int{\left(- \frac{e^{u}}{2}\right)d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=- \frac{1}{2}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\left(- \frac{e^{u}}{2}\right)d u}}} = {\color{red}{\left(- \frac{\int{e^{u} d u}}{2}\right)}}$$

A integral da função exponencial é $$$\int{e^{u} d u} = e^{u}$$$:

$$- \frac{{\color{red}{\int{e^{u} d u}}}}{2} = - \frac{{\color{red}{e^{u}}}}{2}$$

Recorde que $$$u=- x^{2}$$$:

$$- \frac{e^{{\color{red}{u}}}}{2} = - \frac{e^{{\color{red}{\left(- x^{2}\right)}}}}{2}$$

Portanto,

$$\int{x e^{- x^{2}} d x} = - \frac{e^{- x^{2}}}{2}$$

Adicione a constante de integração:

$$\int{x e^{- x^{2}} d x} = - \frac{e^{- x^{2}}}{2}+C$$

$$$\int x e^{- x^{2}}\, dx = - \frac{e^{- x^{2}}}{2} + C$$$A