Integral de $$$70 e^{\frac{3 x}{50}}$$$

Encontre $$$\int 70 e^{\frac{3 x}{50}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=70$$$ e $$$f{\left(x \right)} = e^{\frac{3 x}{50}}$$$:

$${\color{red}{\int{70 e^{\frac{3 x}{50}} d x}}} = {\color{red}{\left(70 \int{e^{\frac{3 x}{50}} d x}\right)}}$$

Seja $$$u=\frac{3 x}{50}$$$.

Então $$$du=\left(\frac{3 x}{50}\right)^{\prime }dx = \frac{3 dx}{50}$$$ (veja os passos »), e obtemos $$$dx = \frac{50 du}{3}$$$.

Assim,

$$70 {\color{red}{\int{e^{\frac{3 x}{50}} d x}}} = 70 {\color{red}{\int{\frac{50 e^{u}}{3} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{50}{3}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$$70 {\color{red}{\int{\frac{50 e^{u}}{3} d u}}} = 70 {\color{red}{\left(\frac{50 \int{e^{u} d u}}{3}\right)}}$$

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

$$\frac{3500 {\color{red}{\int{e^{u} d u}}}}{3} = \frac{3500 {\color{red}{e^{u}}}}{3}$$

Recorde que $$$u=\frac{3 x}{50}$$$:

$$\frac{3500 e^{{\color{red}{u}}}}{3} = \frac{3500 e^{{\color{red}{\left(\frac{3 x}{50}\right)}}}}{3}$$

Portanto,

$$\int{70 e^{\frac{3 x}{50}} d x} = \frac{3500 e^{\frac{3 x}{50}}}{3}$$

Adicione a constante de integração:

$$\int{70 e^{\frac{3 x}{50}} d x} = \frac{3500 e^{\frac{3 x}{50}}}{3}+C$$

$$$\int 70 e^{\frac{3 x}{50}}\, dx = \frac{3500 e^{\frac{3 x}{50}}}{3} + C$$$A