Integral de $$$\frac{1}{\sin^{2}{\left(\frac{x}{3} \right)}}$$$

Encontre $$$\int \frac{1}{\sin^{2}{\left(\frac{x}{3} \right)}}\, dx$$$.

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

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

A integral pode ser reescrita como

$${\color{red}{\int{\frac{1}{\sin^{2}{\left(\frac{x}{3} \right)}} d x}}} = {\color{red}{\int{\frac{3}{\sin^{2}{\left(u \right)}} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=3$$$ e $$$f{\left(u \right)} = \frac{1}{\sin^{2}{\left(u \right)}}$$$:

$${\color{red}{\int{\frac{3}{\sin^{2}{\left(u \right)}} d u}}} = {\color{red}{\left(3 \int{\frac{1}{\sin^{2}{\left(u \right)}} d u}\right)}}$$

Reescreva o integrando em termos da cossecante:

$$3 {\color{red}{\int{\frac{1}{\sin^{2}{\left(u \right)}} d u}}} = 3 {\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}$$

A integral de $$$\csc^{2}{\left(u \right)}$$$ é $$$\int{\csc^{2}{\left(u \right)} d u} = - \cot{\left(u \right)}$$$:

$$3 {\color{red}{\int{\csc^{2}{\left(u \right)} d u}}} = 3 {\color{red}{\left(- \cot{\left(u \right)}\right)}}$$

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

$$- 3 \cot{\left({\color{red}{u}} \right)} = - 3 \cot{\left({\color{red}{\left(\frac{x}{3}\right)}} \right)}$$

Portanto,

$$\int{\frac{1}{\sin^{2}{\left(\frac{x}{3} \right)}} d x} = - 3 \cot{\left(\frac{x}{3} \right)}$$

Adicione a constante de integração:

$$\int{\frac{1}{\sin^{2}{\left(\frac{x}{3} \right)}} d x} = - 3 \cot{\left(\frac{x}{3} \right)}+C$$

$$$\int \frac{1}{\sin^{2}{\left(\frac{x}{3} \right)}}\, dx = - 3 \cot{\left(\frac{x}{3} \right)} + C$$$A