Integral de $$$x^{3} e^{- x}$$$

Encontre $$$\int x^{3} e^{- x}\, dx$$$.

Para a integral $$$\int{x^{3} e^{- x} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=x^{3}$$$ e $$$\operatorname{dv}=e^{- x} dx$$$.

Então $$$\operatorname{du}=\left(x^{3}\right)^{\prime }dx=3 x^{2} dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{e^{- x} d x}=- e^{- x}$$$ (os passos podem ser vistos »).

A integral torna-se

$${\color{red}{\int{x^{3} e^{- x} d x}}}={\color{red}{\left(x^{3} \cdot \left(- e^{- x}\right)-\int{\left(- e^{- x}\right) \cdot 3 x^{2} d x}\right)}}={\color{red}{\left(- x^{3} e^{- x} - \int{\left(- 3 x^{2} e^{- x}\right)d x}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=-3$$$ e $$$f{\left(x \right)} = x^{2} e^{- x}$$$:

$$- x^{3} e^{- x} - {\color{red}{\int{\left(- 3 x^{2} e^{- x}\right)d x}}} = - x^{3} e^{- x} - {\color{red}{\left(- 3 \int{x^{2} e^{- x} d x}\right)}}$$

Para a integral $$$\int{x^{2} e^{- x} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=x^{2}$$$ e $$$\operatorname{dv}=e^{- x} dx$$$.

Então $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{e^{- x} d x}=- e^{- x}$$$ (os passos podem ser vistos »).

Assim,

$$- x^{3} e^{- x} + 3 {\color{red}{\int{x^{2} e^{- x} d x}}}=- x^{3} e^{- x} + 3 {\color{red}{\left(x^{2} \cdot \left(- e^{- x}\right)-\int{\left(- e^{- x}\right) \cdot 2 x d x}\right)}}=- x^{3} e^{- x} + 3 {\color{red}{\left(- x^{2} e^{- x} - \int{\left(- 2 x e^{- x}\right)d x}\right)}}$$

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

$$- x^{3} e^{- x} - 3 x^{2} e^{- x} - 3 {\color{red}{\int{\left(- 2 x e^{- x}\right)d x}}} = - x^{3} e^{- x} - 3 x^{2} e^{- x} - 3 {\color{red}{\left(- 2 \int{x e^{- x} d x}\right)}}$$

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

Sejam $$$\operatorname{u}=x$$$ e $$$\operatorname{dv}=e^{- x} dx$$$.

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

Assim,

$$- x^{3} e^{- x} - 3 x^{2} e^{- x} + 6 {\color{red}{\int{x e^{- x} d x}}}=- x^{3} e^{- x} - 3 x^{2} e^{- x} + 6 {\color{red}{\left(x \cdot \left(- e^{- x}\right)-\int{\left(- e^{- x}\right) \cdot 1 d x}\right)}}=- x^{3} e^{- x} - 3 x^{2} e^{- x} + 6 {\color{red}{\left(- x e^{- x} - \int{\left(- e^{- x}\right)d x}\right)}}$$

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

$$- x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} - 6 {\color{red}{\int{\left(- e^{- x}\right)d x}}} = - x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} - 6 {\color{red}{\left(- \int{e^{- x} d x}\right)}}$$

Seja $$$u=- x$$$.

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

A integral torna-se

$$- x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} + 6 {\color{red}{\int{e^{- x} d x}}} = - x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} + 6 {\color{red}{\int{\left(- e^{u}\right)d u}}}$$

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

$$- x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} + 6 {\color{red}{\int{\left(- e^{u}\right)d u}}} = - x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} + 6 {\color{red}{\left(- \int{e^{u} d u}\right)}}$$

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

$$- x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} - 6 {\color{red}{\int{e^{u} d u}}} = - x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} - 6 {\color{red}{e^{u}}}$$

Recorde que $$$u=- x$$$:

$$- x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} - 6 e^{{\color{red}{u}}} = - x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} - 6 e^{{\color{red}{\left(- x\right)}}}$$

Portanto,

$$\int{x^{3} e^{- x} d x} = - x^{3} e^{- x} - 3 x^{2} e^{- x} - 6 x e^{- x} - 6 e^{- x}$$

Simplifique:

$$\int{x^{3} e^{- x} d x} = \left(- x^{3} - 3 x^{2} - 6 x - 6\right) e^{- x}$$

Adicione a constante de integração:

$$\int{x^{3} e^{- x} d x} = \left(- x^{3} - 3 x^{2} - 6 x - 6\right) e^{- x}+C$$

$$$\int x^{3} e^{- x}\, dx = \left(- x^{3} - 3 x^{2} - 6 x - 6\right) e^{- x} + C$$$A