Integral of cosec(2x) dx

1. Rewrite the integrand:

   $$\csc{\left(2 x \right)} = \frac{\cot{\left(2 x \right)} \csc{\left(2 x \right)} + \csc^{2}{\left(2 x \right)}}{\cot{\left(2 x \right)} + \csc{\left(2 x \right)}}$$

2. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \cot{\left(2 x \right)} + \csc{\left(2 x \right)}$.

      Then let $du = \left(- 2 \cot^{2}{\left(2 x \right)} - 2 \cot{\left(2 x \right)} \csc{\left(2 x \right)} - 2\right) dx$ and substitute $- \frac{du}{2}$:

      $$\int \frac{1}{4 u}\, du$$

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

         $$\int \left(- \frac{1}{2 u}\right)\, du = - \frac{\int \frac{1}{u}\, du}{2}$$

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

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

      Now substitute $u$ back in:

      $$- \frac{\log{\left(\cot{\left(2 x \right)} + \csc{\left(2 x \right)} \right)}}{2}$$

   Method #2

   1. Let $u = 2 x$.

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

      $$\int \frac{\csc{\left(u \right)}}{4}\, du$$

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

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

         1. Rewrite the integrand:

            $$\csc{\left(u \right)} = \frac{\cot{\left(u \right)} \csc{\left(u \right)} + \csc^{2}{\left(u \right)}}{\cot{\left(u \right)} + \csc{\left(u \right)}}$$

         2. Let $u = \cot{\left(u \right)} + \csc{\left(u \right)}$.

            Then let $du = \left(- \cot^{2}{\left(u \right)} - \cot{\left(u \right)} \csc{\left(u \right)} - 1\right) du$ and substitute $- du$:

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

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

               $$\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$$

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

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

            Now substitute $u$ back in:

            $$- \log{\left(\cot{\left(u \right)} + \csc{\left(u \right)} \right)}$$

         So, the result is: $- \frac{\log{\left(\cot{\left(u \right)} + \csc{\left(u \right)} \right)}}{2}$

      Now substitute $u$ back in:

      $$- \frac{\log{\left(\cot{\left(2 x \right)} + \csc{\left(2 x \right)} \right)}}{2}$$

3. Now simplify:

   $$- \frac{\log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{2}$$

4. Add the constant of integration:

   $$- \frac{\log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{2}+ \mathrm{constant}$$

The answer is:

$$- \frac{\log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{2}+ \mathrm{constant}$$