Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 460: 30


$$-\frac{1}{3}\csc ^3\left(x\right)+\csc \left(x\right)+C$$

Work Step by Step

\begin{aligned} \int \cot ^{3} x \csc x d x &=\int \cot ^{2} x \cot x \csc x d x \\ &=\int\left(\csc ^{2} x-1\right) \cot x \csc x d x \\ &=\int \csc ^{3} x \cot x-\cot x \csc x d x \\ &=\int \csc ^{3} x \cot x d x-\int \cot x \csc x d x\\ &=-\frac{1}{3}\csc ^3\left(x\right)+\csc \left(x\right)+C \end{aligned}
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