University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 8 - Section 8.2 - Trigonometric Integrals - Exercises - Page 435: 59


$$\int\cos^3\theta\sin2\theta d\theta=-\frac{2}{5}\cos^5\theta+C$$

Work Step by Step

$$A=\int\cos^3\theta\sin2\theta d\theta$$ Use the identity: $$\sin2\theta=2\sin\theta\cos\theta$$ we have $$A=\int\cos^3\theta(2\sin\theta\cos\theta)d\theta$$ $$A=2\int\cos^4\theta\sin\theta d\theta$$ $$A=-2\int\cos^4\theta d(\cos\theta)$$ We set $u=\cos\theta$ $$A=-2\int u^4du=-\frac{2}{5}u^5+C$$ $$A=-\frac{2}{5}\cos^5\theta+C$$
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