Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.2 Exercises - Page 334: 33

$-\frac{1}{2}ln|csc(2x)+cot(2x)| +C$

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