Integral de $$$2 e^{2 x}$$$

Encontre $$$\int 2 e^{2 x}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=2$$$ e $$$f{\left(x \right)} = e^{2 x}$$$:

$${\color{red}{\int{2 e^{2 x} d x}}} = {\color{red}{\left(2 \int{e^{2 x} d x}\right)}}$$

Seja $$$u=2 x$$$.

Então $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{2}$$$.

Portanto,

$$2 {\color{red}{\int{e^{2 x} d x}}} = 2 {\color{red}{\int{\frac{e^{u}}{2} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$$2 {\color{red}{\int{\frac{e^{u}}{2} d u}}} = 2 {\color{red}{\left(\frac{\int{e^{u} d u}}{2}\right)}}$$

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 $$$u=2 x$$$:

$$e^{{\color{red}{u}}} = e^{{\color{red}{\left(2 x\right)}}}$$

Portanto,

$$\int{2 e^{2 x} d x} = e^{2 x}$$

Adicione a constante de integração:

$$\int{2 e^{2 x} d x} = e^{2 x}+C$$

$$$\int 2 e^{2 x}\, dx = e^{2 x} + C$$$A