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

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

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 »), and we have that $\cos{\left(x \right)} dx = du$.

Thus,

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

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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