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

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

Let $$$u=2 x$$$.

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

The integral can be rewritten as

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

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

$${\color{red}{\int{\frac{\cot{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\cot{\left(u \right)} d u}}{2}\right)}}$$

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

$$\frac{{\color{red}{\int{\cot{\left(u \right)} d u}}}}{2} = \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}}{2}$$

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

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

Therefore,

$$\frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}}{2} = \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{2}$$

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

$$\frac{{\color{red}{\int{\frac{1}{v} d v}}}}{2} = \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{2}$$

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

$$\frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{2} = \frac{\ln{\left(\left|{{\color{red}{\sin{\left(u \right)}}}}\right| \right)}}{2}$$

Recall that $$$u=2 x$$$:

$$\frac{\ln{\left(\left|{\sin{\left({\color{red}{u}} \right)}}\right| \right)}}{2} = \frac{\ln{\left(\left|{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}\right| \right)}}{2}$$

Therefore,

$$\int{\cot{\left(2 x \right)} d x} = \frac{\ln{\left(\left|{\sin{\left(2 x \right)}}\right| \right)}}{2}$$

Add the constant of integration:

$$\int{\cot{\left(2 x \right)} d x} = \frac{\ln{\left(\left|{\sin{\left(2 x \right)}}\right| \right)}}{2}+C$$

$$$\int \cot{\left(2 x \right)}\, dx = \frac{\ln\left(\left|{\sin{\left(2 x \right)}}\right|\right)}{2} + C$$$A