Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - 7.2 Trigonometric Integrals - 7.2 Exercises: 2

Answer

$\frac{1}{7}\cos^7\theta-\frac{1}{5}\cos^5\theta+C$

Work Step by Step

$\int \sin^3\theta\cos^4\theta\ d\theta$ Since the power of sine is odd, save a factor of $\sin\theta$ and express the rest in terms of $\cos \theta$: $=\int\sin^2\theta\sin\theta\cos^4\theta\ d\theta$ $=\int(1-\cos^2\theta)\cos^4\theta\sin\theta\ d\theta$ Let $u=\cos\theta$. Then $du=-\sin\theta\ d\theta$, and $\sin\theta\ d\theta=-du$ $=\int(1-u^2)u^4*(-1)du$ $=\int(u^4-u^6)*(-1)du$ $=\int(u^6-u^4)du$ $=\frac{1}{7}u^7-\frac{1}{5}u^5+C$ $=\boxed{\frac{1}{7}\cos^7\theta-\frac{1}{5}\cos^5\theta+C}$
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