Integral de $$$\csc^{2}{\left(4 x \right)}$$$

Encontre $$$\int \csc^{2}{\left(4 x \right)}\, dx$$$.

Seja $$$u=4 x$$$.

Então $$$du=\left(4 x\right)^{\prime }dx = 4 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{4}$$$.

Logo,

$${\color{red}{\int{\csc^{2}{\left(4 x \right)} d x}}} = {\color{red}{\int{\frac{\csc^{2}{\left(u \right)}}{4} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{4}$$$ e $$$f{\left(u \right)} = \csc^{2}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\csc^{2}{\left(u \right)}}{4} d u}}} = {\color{red}{\left(\frac{\int{\csc^{2}{\left(u \right)} d u}}{4}\right)}}$$

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

$$\frac{{\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}}{4} = \frac{{\color{red}{\left(- \cot{\left(u \right)}\right)}}}{4}$$

Recorde que $$$u=4 x$$$:

$$- \frac{\cot{\left({\color{red}{u}} \right)}}{4} = - \frac{\cot{\left({\color{red}{\left(4 x\right)}} \right)}}{4}$$

Portanto,

$$\int{\csc^{2}{\left(4 x \right)} d x} = - \frac{\cot{\left(4 x \right)}}{4}$$

Adicione a constante de integração:

$$\int{\csc^{2}{\left(4 x \right)} d x} = - \frac{\cot{\left(4 x \right)}}{4}+C$$

$$$\int \csc^{2}{\left(4 x \right)}\, dx = - \frac{\cot{\left(4 x \right)}}{4} + C$$$A