Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.2 Exercises - Page 334: 34

$2ln|sec(\frac{1}{2}x)+tan(\frac{1}{2}x)|+C$

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