Integral de $$$\sin{\left(\frac{3 u}{5} \right)}$$$

Encontre $$$\int \sin{\left(\frac{3 u}{5} \right)}\, du$$$.

Seja $$$v=\frac{3 u}{5}$$$.

Então $$$dv=\left(\frac{3 u}{5}\right)^{\prime }du = \frac{3 du}{5}$$$ (veja os passos »), e obtemos $$$du = \frac{5 dv}{3}$$$.

A integral torna-se

$${\color{red}{\int{\sin{\left(\frac{3 u}{5} \right)} d u}}} = {\color{red}{\int{\frac{5 \sin{\left(v \right)}}{3} d v}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ usando $$$c=\frac{5}{3}$$$ e $$$f{\left(v \right)} = \sin{\left(v \right)}$$$:

$${\color{red}{\int{\frac{5 \sin{\left(v \right)}}{3} d v}}} = {\color{red}{\left(\frac{5 \int{\sin{\left(v \right)} d v}}{3}\right)}}$$

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

$$\frac{5 {\color{red}{\int{\sin{\left(v \right)} d v}}}}{3} = \frac{5 {\color{red}{\left(- \cos{\left(v \right)}\right)}}}{3}$$

Recorde que $$$v=\frac{3 u}{5}$$$:

$$- \frac{5 \cos{\left({\color{red}{v}} \right)}}{3} = - \frac{5 \cos{\left({\color{red}{\left(\frac{3 u}{5}\right)}} \right)}}{3}$$

Portanto,

$$\int{\sin{\left(\frac{3 u}{5} \right)} d u} = - \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}$$

Adicione a constante de integração:

$$\int{\sin{\left(\frac{3 u}{5} \right)} d u} = - \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}+C$$

$$$\int \sin{\left(\frac{3 u}{5} \right)}\, du = - \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3} + C$$$A