Integral of tg(2x)dx dx

1. Rewrite the integrand:

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

2. There are multiple ways to do this integral.

   Method #1

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

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

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

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

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

         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:

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

   Method #2

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

      $$\int \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(2 x \right)}}\, dx = 2 \int \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(2 x \right)}}\, dx$$

      1. Rewrite the integrand:

         $$\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(2 x \right)}} = \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1}$$

      2. Let $u = 2 \cos^{2}{\left(x \right)} - 1$.

         Then let $du = - 4 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $- \frac{du}{4}$:

         $$\int \left(- \frac{1}{4 u}\right)\, du$$

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

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

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

            So, the result is: $- \frac{\log{\left(u \right)}}{4}$

         Now substitute $u$ back in:

         $$- \frac{\log{\left(2 \cos^{2}{\left(x \right)} - 1 \right)}}{4}$$

      So, the result is: $- \frac{\log{\left(2 \cos^{2}{\left(x \right)} - 1 \right)}}{2}$

3. Add the constant of integration:

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

The answer is:

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