Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 7 - Principles Of Integral Evaluation - 7.3 Integrating Trigonometric Functions - Exercises Set 7.3 - Page 507: 53



Work Step by Step

Changing $\sin^{2}(x)$ to $1-\cos^{2}(x)$ changes the integral to $$\int{\sin(x)(1-\cos^{2})^{2}\cos^{8}(x)}dx$$. Performing a u-substitution with $u=\cos(x)$ and $-du=\sin(x)dx$ leaves $$-\int{u^{8}(1-u^{2})^{2}}du$$ This can be expanded and solved using power rule.
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