Integral of $$$\frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}}$$$

Find $$$\int \frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$$.

Simplify the integrand:

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

Integrate term by term:

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

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

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

Rewrite the integrand in terms of the cosecant:

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

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

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

Therefore,

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

Add the constant of integration:

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

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