Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Section 8.3 - Trigonometric Integrals - Exercises 8.3 - Page 463: 60


$$\frac{2}{5} \cos ^{5} \theta-\cos ^{3} \theta+\cos \theta+C $$

Work Step by Step

We integrate as follows: \begin{align*} \int \sin ^{3} \theta \cos 2 \theta d \theta&=\int \sin ^{2} \theta \cos 2 \theta \sin \theta d \theta\\ &=\int\left(1-\cos ^{2} \theta\right)\left(2 \cos ^{2} \theta-1\right) \sin \theta d \theta\\ &=\int\left(-2 \cos ^{4} \theta+3 \cos ^{2} \theta-1\right) \sin \theta d \theta\\ &=-2 \int \cos ^{4} \theta \sin \theta d \theta+3 \int \cos ^{2} \theta \sin \theta d \theta-\int \sin \theta d \theta\\ &=\frac{2}{5} \cos ^{5} \theta-\cos ^{3} \theta+\cos \theta+C \end{align*}
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