Integral de $$$y \sin{\left(2 x \right)}$$$ em relação a $$$x$$$

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

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

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

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

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

Assim,

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

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 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

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

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

Recorde que $$$u=2 x$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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