Integral of 4sec(2x)tan(2x)dx dx

1. Let $u = 2 x$.

   Then let $du = 2 dx$ and substitute $2 du$:

   $$\int 2 \tan{\left(u \right)} \sec{\left(u \right)}\, du$$

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

      $$\int \tan{\left(u \right)} \sec{\left(u \right)}\, du = 2 \int \tan{\left(u \right)} \sec{\left(u \right)}\, du$$

      1. The integral of secant times tangent is secant:

         $$\int \tan{\left(u \right)} \sec{\left(u \right)}\, du = \sec{\left(u \right)}$$

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

   Now substitute $u$ back in:

   $$2 \sec{\left(2 x \right)}$$

2. Add the constant of integration:

   $$2 \sec{\left(2 x \right)}+ \mathrm{constant}$$

The answer is:

$$2 \sec{\left(2 x \right)}+ \mathrm{constant}$$