Integral of tg(4*x) dx

1. Rewrite the integrand:

   $$\tan{\left(4 x \right)} = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$$

2. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \cos{\left(4 x \right)}$.

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

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

   Method #2

   1. Let $u = 4 x$.

      Then let $du = 4 dx$ and substitute $\frac{du}{4}$:

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

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

         $$\int \frac{\sin{\left(u \right)}}{\cos{\left(u \right)}}\, du = \frac{\int \frac{\sin{\left(u \right)}}{\cos{\left(u \right)}}\, du}{4}$$

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

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

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

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

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

               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:

            $$- \log{\left(\cos{\left(u \right)} \right)}$$

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

      Now substitute $u$ back in:

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

3. Add the constant of integration:

   $$- \frac{\log{\left(\cos{\left(4 x \right)} \right)}}{4}+ \mathrm{constant}$$

The answer is: