Integral de $$$e^{- 4 x}$$$

Encontre $$$\int e^{- 4 x}\, dx$$$.

Seja $$$u=- 4 x$$$.

Então $$$du=\left(- 4 x\right)^{\prime }dx = - 4 dx$$$ (veja os passos »), e obtemos $$$dx = - \frac{du}{4}$$$.

Assim,

$${\color{red}{\int{e^{- 4 x} d x}}} = {\color{red}{\int{\left(- \frac{e^{u}}{4}\right)d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=- \frac{1}{4}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\left(- \frac{e^{u}}{4}\right)d u}}} = {\color{red}{\left(- \frac{\int{e^{u} d u}}{4}\right)}}$$

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

$$- \frac{{\color{red}{\int{e^{u} d u}}}}{4} = - \frac{{\color{red}{e^{u}}}}{4}$$

Recorde que $$$u=- 4 x$$$:

$$- \frac{e^{{\color{red}{u}}}}{4} = - \frac{e^{{\color{red}{\left(- 4 x\right)}}}}{4}$$

Portanto,

$$\int{e^{- 4 x} d x} = - \frac{e^{- 4 x}}{4}$$

Adicione a constante de integração:

$$\int{e^{- 4 x} d x} = - \frac{e^{- 4 x}}{4}+C$$

$$$\int e^{- 4 x}\, dx = - \frac{e^{- 4 x}}{4} + C$$$A