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 $$$dx = \frac{du}{4}$$$.

Portanto,

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

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 $$$f{\left(u \right)} = e^{u}$$$:

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

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