Integral de $$$\frac{\csc^{2}{\left(x \right)}}{9}$$$

Encontre $$$\int \frac{\csc^{2}{\left(x \right)}}{9}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{9}$$$ e $$$f{\left(x \right)} = \csc^{2}{\left(x \right)}$$$:

$${\color{red}{\int{\frac{\csc^{2}{\left(x \right)}}{9} d x}}} = {\color{red}{\left(\frac{\int{\csc^{2}{\left(x \right)} d x}}{9}\right)}}$$

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

$$\frac{{\color{red}{\int{\csc^{2}{\left(x \right)} d x}}}}{9} = \frac{{\color{red}{\left(- \cot{\left(x \right)}\right)}}}{9}$$

Portanto,

$$\int{\frac{\csc^{2}{\left(x \right)}}{9} d x} = - \frac{\cot{\left(x \right)}}{9}$$

Adicione a constante de integração:

$$\int{\frac{\csc^{2}{\left(x \right)}}{9} d x} = - \frac{\cot{\left(x \right)}}{9}+C$$

$$$\int \frac{\csc^{2}{\left(x \right)}}{9}\, dx = - \frac{\cot{\left(x \right)}}{9} + C$$$A