Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 460: 31

$$-\frac{1}{3} \cot ^{3} x+C$$

Given $$ \int \cot ^{2} x \csc ^{2} x d x $$ Let $ u=\cot x\ \ \to \ \ du=-\csc ^{2} x d x $ \begin{aligned} \int \cot ^{2} x \csc ^{2} x d x &=-\int u^{2} d u \\ &=-\frac{u^{3}}{3}+C \\ &=-\frac{1}{3} \cot ^{3} x+C \end{aligned}