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

The calculator will find the integral/antiderivative of $\cot^{2}{\left(x \right)}$, with steps shown.

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

### Solution

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,

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

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