Integral de $$$220 e^{\frac{x}{10}}$$$

Encontre $$$\int 220 e^{\frac{x}{10}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=220$$$ e $$$f{\left(x \right)} = e^{\frac{x}{10}}$$$:

$${\color{red}{\int{220 e^{\frac{x}{10}} d x}}} = {\color{red}{\left(220 \int{e^{\frac{x}{10}} d x}\right)}}$$

Seja $$$u=\frac{x}{10}$$$.

Então $$$du=\left(\frac{x}{10}\right)^{\prime }dx = \frac{dx}{10}$$$ (veja os passos »), e obtemos $$$dx = 10 du$$$.

A integral torna-se

$$220 {\color{red}{\int{e^{\frac{x}{10}} d x}}} = 220 {\color{red}{\int{10 e^{u} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=10$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$$220 {\color{red}{\int{10 e^{u} d u}}} = 220 {\color{red}{\left(10 \int{e^{u} d u}\right)}}$$

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

$$2200 {\color{red}{\int{e^{u} d u}}} = 2200 {\color{red}{e^{u}}}$$

Recorde que $$$u=\frac{x}{10}$$$:

$$2200 e^{{\color{red}{u}}} = 2200 e^{{\color{red}{\left(\frac{x}{10}\right)}}}$$

Portanto,

$$\int{220 e^{\frac{x}{10}} d x} = 2200 e^{\frac{x}{10}}$$

Adicione a constante de integração:

$$\int{220 e^{\frac{x}{10}} d x} = 2200 e^{\frac{x}{10}}+C$$

$$$\int 220 e^{\frac{x}{10}}\, dx = 2200 e^{\frac{x}{10}} + C$$$A