Integral de $$$e^{- 6 x}$$$

Encontre $$$\int e^{- 6 x}\, dx$$$.

Seja $$$u=- 6 x$$$.

Então $$$du=\left(- 6 x\right)^{\prime }dx = - 6 dx$$$ (veja os passos »), e obtemos $$$dx = - \frac{du}{6}$$$.

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=- \frac{1}{6}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\left(- \frac{e^{u}}{6}\right)d u}}} = {\color{red}{\left(- \frac{\int{e^{u} d u}}{6}\right)}}$$

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

$$- \frac{{\color{red}{\int{e^{u} d u}}}}{6} = - \frac{{\color{red}{e^{u}}}}{6}$$

Recorde que $$$u=- 6 x$$$:

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

Portanto,

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

Adicione a constante de integração:

$$\int{e^{- 6 x} d x} = - \frac{e^{- 6 x}}{6}+C$$

$$$\int e^{- 6 x}\, dx = - \frac{e^{- 6 x}}{6} + C$$$A