Integral de $$$x e^{- 5 x}$$$

Encontre $$$\int x e^{- 5 x}\, dx$$$.

Para a integral $$$\int{x e^{- 5 x} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=x$$$ e $$$\operatorname{dv}=e^{- 5 x} dx$$$.

Então $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{e^{- 5 x} d x}=- \frac{e^{- 5 x}}{5}$$$ (os passos podem ser vistos »).

A integral pode ser reescrita como

$${\color{red}{\int{x e^{- 5 x} d x}}}={\color{red}{\left(x \cdot \left(- \frac{e^{- 5 x}}{5}\right)-\int{\left(- \frac{e^{- 5 x}}{5}\right) \cdot 1 d x}\right)}}={\color{red}{\left(- \frac{x e^{- 5 x}}{5} - \int{\left(- \frac{e^{- 5 x}}{5}\right)d x}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=- \frac{1}{5}$$$ e $$$f{\left(x \right)} = e^{- 5 x}$$$:

$$- \frac{x e^{- 5 x}}{5} - {\color{red}{\int{\left(- \frac{e^{- 5 x}}{5}\right)d x}}} = - \frac{x e^{- 5 x}}{5} - {\color{red}{\left(- \frac{\int{e^{- 5 x} d x}}{5}\right)}}$$

Seja $$$u=- 5 x$$$.

Então $$$du=\left(- 5 x\right)^{\prime }dx = - 5 dx$$$ (veja os passos »), e obtemos $$$dx = - \frac{du}{5}$$$.

A integral torna-se

$$- \frac{x e^{- 5 x}}{5} + \frac{{\color{red}{\int{e^{- 5 x} d x}}}}{5} = - \frac{x e^{- 5 x}}{5} + \frac{{\color{red}{\int{\left(- \frac{e^{u}}{5}\right)d u}}}}{5}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=- \frac{1}{5}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$$- \frac{x e^{- 5 x}}{5} + \frac{{\color{red}{\int{\left(- \frac{e^{u}}{5}\right)d u}}}}{5} = - \frac{x e^{- 5 x}}{5} + \frac{{\color{red}{\left(- \frac{\int{e^{u} d u}}{5}\right)}}}{5}$$

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

$$- \frac{x e^{- 5 x}}{5} - \frac{{\color{red}{\int{e^{u} d u}}}}{25} = - \frac{x e^{- 5 x}}{5} - \frac{{\color{red}{e^{u}}}}{25}$$

Recorde que $$$u=- 5 x$$$:

$$- \frac{x e^{- 5 x}}{5} - \frac{e^{{\color{red}{u}}}}{25} = - \frac{x e^{- 5 x}}{5} - \frac{e^{{\color{red}{\left(- 5 x\right)}}}}{25}$$

Portanto,

$$\int{x e^{- 5 x} d x} = - \frac{x e^{- 5 x}}{5} - \frac{e^{- 5 x}}{25}$$

Simplifique:

$$\int{x e^{- 5 x} d x} = \frac{\left(- 5 x - 1\right) e^{- 5 x}}{25}$$

Adicione a constante de integração:

$$\int{x e^{- 5 x} d x} = \frac{\left(- 5 x - 1\right) e^{- 5 x}}{25}+C$$

$$$\int x e^{- 5 x}\, dx = \frac{\left(- 5 x - 1\right) e^{- 5 x}}{25} + C$$$A