Integral de $$$t^{3} e^{- t^{2}}$$$

Encontre $$$\int t^{3} e^{- t^{2}}\, dt$$$.

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

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

Portanto,

$${\color{red}{\int{t^{3} e^{- t^{2}} d t}}} = {\color{red}{\int{\frac{u e^{u}}{2} d u}}}$$

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 $$$f{\left(u \right)} = u e^{u}$$$:

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

Para a integral $$$\int{u e^{u} d u}$$$, use integração por partes $$$\int \operatorname{c} \operatorname{dv} = \operatorname{c}\operatorname{v} - \int \operatorname{v} \operatorname{dc}$$$.

Sejam $$$\operatorname{c}=u$$$ e $$$\operatorname{dv}=e^{u} du$$$.

Então $$$\operatorname{dc}=\left(u\right)^{\prime }du=1 du$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{e^{u} d u}=e^{u}$$$ (os passos podem ser vistos »).

A integral torna-se

$$\frac{{\color{red}{\int{u e^{u} d u}}}}{2}=\frac{{\color{red}{\left(u \cdot e^{u}-\int{e^{u} \cdot 1 d u}\right)}}}{2}=\frac{{\color{red}{\left(u e^{u} - \int{e^{u} d u}\right)}}}{2}$$

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

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

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

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

Portanto,

$$\int{t^{3} e^{- t^{2}} d t} = - \frac{t^{2} e^{- t^{2}}}{2} - \frac{e^{- t^{2}}}{2}$$

Simplifique:

$$\int{t^{3} e^{- t^{2}} d t} = \frac{\left(- t^{2} - 1\right) e^{- t^{2}}}{2}$$

Adicione a constante de integração:

$$\int{t^{3} e^{- t^{2}} d t} = \frac{\left(- t^{2} - 1\right) e^{- t^{2}}}{2}+C$$

$$$\int t^{3} e^{- t^{2}}\, dt = \frac{\left(- t^{2} - 1\right) e^{- t^{2}}}{2} + C$$$A