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

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

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### Your Input

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

### Solution

Rewrite the cotangent as $\cot\left(x\right)=\frac{\cos\left(x\right)}{\sin\left(x\right)}$:

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

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

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

The integral becomes

$$\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}} = \color{red}{\int{\frac{1}{u} d u}}$$

The integral of $\frac{1}{u}$ is $\int{\frac{1}{u} d u} = \ln{\left(u \right)}$

$$\color{red}{\int{\frac{1}{u} d u}} = \color{red}{\ln{\left(u \right)}}$$

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

$$\ln{\left(\color{red}{u} \right)} = \ln{\left(\color{red}{\sin{\left(x \right)}} \right)}$$

Therefore,

$$\int{\cot{\left(x \right)} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$

Add the constant of integration:

$$\int{\cot{\left(x \right)} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}+C$$

Answer: $\int{\cot{\left(x \right)} d x}=\ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}+C$