Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 41


$$\frac{-1}{9}\csc^{9} x+\frac{2}{7} \csc ^{7} x-\frac{1}{5}\csc ^{5} x+c$$

Work Step by Step

\begin{align*} \int \cot ^{5} x \csc ^{5} x d x&=\int\left(\csc ^{2} x-1\right)^{2} \csc ^{4} x(\cot x \csc x) d x\\ &=\int\left(\csc ^{8} x-2 \csc ^{6} x+\csc ^{4} x\right)(\cot x \csc x d x) \\ &=\frac{-1}{9}\csc^{9} x+\frac{2}{7} \csc ^{7} x-\frac{1}{5}\csc ^{5} x+c \end{align*}
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