Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 51



Work Step by Step

\begin{align*} \int_{0}^{ \pi/2}\sin^5xdx&= \int_{0}^{ \pi/2}\sin^4x\sin xdx \\ &= \int_{0}^{ \pi/2}(1-\cos^2x)^2\sin xdx \\ &= \int_{0}^{ \pi/2} \left(1-2\cos^2x+\cos^4x\right) \sin xdx \\ &= \left(-\cos x+\frac{2}{3}\cos^3x -\frac{1}{5}\cos^5x \right) \bigg|_{0}^{\pi/2}\\ &=\frac{8}{15} \end{align*}
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