Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 14

$$-x \csc x-\ln |\csc x+\cot x|+C$$

Given $$\int x \csc x \cot x \, d x$$ Let \begin{align*} u&= x\ \ \ \ \ \ dv= \csc x \cot x \, d x\\ du&= dx\ \ \ \ \ \ v=- \csc x \end{align*} Then \begin{align*} \int x \csc x \cot x \, d x&= x\csc x +\int \csc xdx\\ &=-x \csc x-\ln |\csc x+\cot x|+C \end{align*}