Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.2 Exercises - Page 334: 40

$\frac{ln(|tan(2x)+sec(2x)|)-ln(|cos(2x)|)}{2}+C$

$\int (tan(2x)+sec(2x))dx$ $=\int tan(2x)dx+\int sec(2x)dx$ let $2x=u$ $2dx=du$ $=\int tan(2x)dx+\int sec(2x)dx$ $=\frac{1}{2} \int tan(u)du + \frac{1}{2} \int sec(u)du$ $=(\frac{-ln(|cos(u)|)}{2})+(\frac{ln(|tan(u)+sec(u)|)}{2})+C$ $=\frac{ln(|tan(2x)+sec(2x)|)-ln(|cos(2x)|)}{2}+C$