Integral de $$$- 7 e^{- 7 x}$$$

Encontre $$$\int \left(- 7 e^{- 7 x}\right)\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=-7$$$ e $$$f{\left(x \right)} = e^{- 7 x}$$$:

$${\color{red}{\int{\left(- 7 e^{- 7 x}\right)d x}}} = {\color{red}{\left(- 7 \int{e^{- 7 x} d x}\right)}}$$

Seja $$$u=- 7 x$$$.

Então $$$du=\left(- 7 x\right)^{\prime }dx = - 7 dx$$$ (veja os passos »), e obtemos $$$dx = - \frac{du}{7}$$$.

Logo,

$$- 7 {\color{red}{\int{e^{- 7 x} d x}}} = - 7 {\color{red}{\int{\left(- \frac{e^{u}}{7}\right)d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=- \frac{1}{7}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$$- 7 {\color{red}{\int{\left(- \frac{e^{u}}{7}\right)d u}}} = - 7 {\color{red}{\left(- \frac{\int{e^{u} d u}}{7}\right)}}$$

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

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

Recorde que $$$u=- 7 x$$$:

$$e^{{\color{red}{u}}} = e^{{\color{red}{\left(- 7 x\right)}}}$$

Portanto,

$$\int{\left(- 7 e^{- 7 x}\right)d x} = e^{- 7 x}$$

Adicione a constante de integração:

$$\int{\left(- 7 e^{- 7 x}\right)d x} = e^{- 7 x}+C$$

$$$\int \left(- 7 e^{- 7 x}\right)\, dx = e^{- 7 x} + C$$$A