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 459: 3


$$\frac{1}{9} \sin ^{9} \theta-\frac{1}{11} \sin ^{11} \theta+C$$

Work Step by Step

\begin{aligned} \int \cos ^{3} \theta \sin ^{8} \theta d \theta &=\int \cos \theta \cos ^{2} \theta \sin ^{8} \theta d \theta \\ &=\int \cos \theta\left(1-\sin ^{2} \theta\right) \sin ^{8} \theta d \theta \\ &=\int\left(\cos \theta-\cos \theta \sin ^{2} \theta\right) \sin ^{8} \theta d \theta \\ &=\int\left(\cos \theta \sin ^{8} \theta-\cos \theta \sin ^{10} \theta\right) d \theta \\ &=\int \cos \theta \sin ^{8} \theta d \theta-\int \cos \theta \sin ^{10} \theta d \theta\\ &=\frac{1}{9} \sin ^{9} \theta-\frac{1}{11} \sin ^{11} \theta+C \end{aligned}
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