Integral of $$$\cot^{2}{\left(x \right)}$$$

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

Let $$$u=\cot{\left(x \right)}$$$.

Then $$$du=\left(\cot{\left(x \right)}\right)^{\prime }dx = - \csc^{2}{\left(x \right)} dx$$$ (steps can be seen here), and we have that $$$\csc^{2}{\left(x \right)} dx = - du$$$.

The integral can be rewritten as

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \frac{u^{2}}{u^{2} + 1}$$$:

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

Rewrite and split the fraction:

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

Integrate term by term:

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

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=1$$$:

$$\int{\frac{1}{u^{2} + 1} d u} - \color{red}{\int{1 d u}} = \int{\frac{1}{u^{2} + 1} d u} - \color{red}{u}$$

The integral of $$$\frac{1}{u^{2} + 1}$$$ is $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:

$$- u + \color{red}{\int{\frac{1}{u^{2} + 1} d u}} = - u + \color{red}{\operatorname{atan}{\left(u \right)}}$$

Recall that $$$u=\cot{\left(x \right)}$$$:

$$\operatorname{atan}{\left(\color{red}{u} \right)} - \color{red}{u} = \operatorname{atan}{\left(\color{red}{\cot{\left(x \right)}} \right)} - \color{red}{\cot{\left(x \right)}}$$

Therefore,

$$\int{\cot^{2}{\left(x \right)} d x} = - \cot{\left(x \right)} + \operatorname{atan}{\left(\cot{\left(x \right)} \right)}$$

Add the constant of integration:

$$\int{\cot^{2}{\left(x \right)} d x} = - \cot{\left(x \right)} + \operatorname{atan}{\left(\cot{\left(x \right)} \right)}+C$$

Answer: $$$\int{\cot^{2}{\left(x \right)} d x}=- \cot{\left(x \right)} + \operatorname{atan}{\left(\cot{\left(x \right)} \right)}+C$$$