Integral of ctg(x/2) dx

1. Rewrite the integrand:

   $$\cot{\left(\frac{x}{2} \right)} = \frac{\cos{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}}$$

2. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \sin{\left(\frac{x}{2} \right)}$.

      Then let $du = \frac{\cos{\left(\frac{x}{2} \right)} dx}{2}$ and substitute $2 du$:

      $$\int \frac{2}{u}\, du$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \frac{1}{u}\, du = 2 \int \frac{1}{u}\, du$$

         1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

         So, the result is: $2 \log{\left(u \right)}$

      Now substitute $u$ back in:

      $$2 \log{\left(\sin{\left(\frac{x}{2} \right)} \right)}$$

   Method #2

   1. Let $u = \frac{x}{2}$.

      Then let $du = \frac{dx}{2}$ and substitute $2 du$:

      $$\int \frac{2 \cos{\left(u \right)}}{\sin{\left(u \right)}}\, du$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}\, du = 2 \int \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}\, du$$

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

            Then let $du = \cos{\left(u \right)} du$ and substitute $du$:

            $$\int \frac{1}{u}\, du$$

            1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

            Now substitute $u$ back in:

            $$\log{\left(\sin{\left(u \right)} \right)}$$

         So, the result is: $2 \log{\left(\sin{\left(u \right)} \right)}$

      Now substitute $u$ back in:

      $$2 \log{\left(\sin{\left(\frac{x}{2} \right)} \right)}$$

3. Now simplify:

   $$2 \log{\left(\sin{\left(\frac{x}{2} \right)} \right)}$$

4. Add the constant of integration:

   $$2 \log{\left(\sin{\left(\frac{x}{2} \right)} \right)}+ \mathrm{constant}$$

The answer is:

$$2 \log{\left(\sin{\left(\frac{x}{2} \right)} \right)}+ \mathrm{constant}$$