Integral de $$$\sin{\left(\frac{x}{2} \right)}$$$

Encontre $$$\int \sin{\left(\frac{x}{2} \right)}\, dx$$$.

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

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

A integral pode ser reescrita como

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

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

$${\color{red}{\int{2 \sin{\left(u \right)} d u}}} = {\color{red}{\left(2 \int{\sin{\left(u \right)} d u}\right)}}$$

A integral do seno é $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$2 {\color{red}{\int{\sin{\left(u \right)} d u}}} = 2 {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$

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

$$- 2 \cos{\left({\color{red}{u}} \right)} = - 2 \cos{\left({\color{red}{\left(\frac{x}{2}\right)}} \right)}$$

Portanto,

$$\int{\sin{\left(\frac{x}{2} \right)} d x} = - 2 \cos{\left(\frac{x}{2} \right)}$$

Adicione a constante de integração:

$$\int{\sin{\left(\frac{x}{2} \right)} d x} = - 2 \cos{\left(\frac{x}{2} \right)}+C$$

$$$\int \sin{\left(\frac{x}{2} \right)}\, dx = - 2 \cos{\left(\frac{x}{2} \right)} + C$$$A