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

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

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

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

Assim,

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

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

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

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

$$$\int e^{- 5 x}\, dx = - \frac{e^{- 5 x}}{5} + C$$$A