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

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

Rewrite the integrand in terms of the cosecant:

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

The integral of $$$\csc^{2}{\left(x \right)}$$$ is $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

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