Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 47


$$-\frac{1}{3}\cos^3 x+\frac{2}{5}\cos^5x-\frac{1}{7}\cos^7x+C$$

Work Step by Step

\begin{align*} \int \sin ^{5} x \cos ^{2} x d x&=\int \sin ^{4} x \cos ^{2} x \sin xd x\\ &=\int (1-\cos^{2}x)^2 \cos ^{2} x \sin xd x\\ &=\int ( \cos ^{2} x-2\cos^{4}x+\cos^6x)\sin xdx\\ &=-\frac{1}{3}\cos^3 x+\frac{2}{5}\cos^5x-\frac{1}{7}\cos^7x+C \end{align*}
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