Integral de $$$20 e^{\frac{3 x}{2}}$$$

Encontre $$$\int 20 e^{\frac{3 x}{2}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=20$$$ e $$$f{\left(x \right)} = e^{\frac{3 x}{2}}$$$:

$${\color{red}{\int{20 e^{\frac{3 x}{2}} d x}}} = {\color{red}{\left(20 \int{e^{\frac{3 x}{2}} d x}\right)}}$$

Seja $$$u=\frac{3 x}{2}$$$.

Então $$$du=\left(\frac{3 x}{2}\right)^{\prime }dx = \frac{3 dx}{2}$$$ (veja os passos »), e obtemos $$$dx = \frac{2 du}{3}$$$.

Logo,

$$20 {\color{red}{\int{e^{\frac{3 x}{2}} d x}}} = 20 {\color{red}{\int{\frac{2 e^{u}}{3} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{2}{3}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$$20 {\color{red}{\int{\frac{2 e^{u}}{3} d u}}} = 20 {\color{red}{\left(\frac{2 \int{e^{u} d u}}{3}\right)}}$$

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

$$\frac{40 {\color{red}{\int{e^{u} d u}}}}{3} = \frac{40 {\color{red}{e^{u}}}}{3}$$

Recorde que $$$u=\frac{3 x}{2}$$$:

$$\frac{40 e^{{\color{red}{u}}}}{3} = \frac{40 e^{{\color{red}{\left(\frac{3 x}{2}\right)}}}}{3}$$

Portanto,

$$\int{20 e^{\frac{3 x}{2}} d x} = \frac{40 e^{\frac{3 x}{2}}}{3}$$

Adicione a constante de integração:

$$\int{20 e^{\frac{3 x}{2}} d x} = \frac{40 e^{\frac{3 x}{2}}}{3}+C$$

$$$\int 20 e^{\frac{3 x}{2}}\, dx = \frac{40 e^{\frac{3 x}{2}}}{3} + C$$$A