Integral de $$$e^{\frac{x}{4}}$$$

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

Seja $$$u=\frac{x}{4}$$$.

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

A integral torna-se

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

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

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

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

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

Recorde que $$$u=\frac{x}{4}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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