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 404: 74



Work Step by Step

We integrate as follows: \begin{aligned} \int_{0}^{\pi} \sin ^{2} m x d x &=\frac{1}{m} \int_{0}^{m \pi} \sin ^{2} u d u \\ &=\frac{1}{m} \int_{0}^{m \pi}\left(\frac{1-\cos 2 u}{2}\right) d u \\ &=\left.\frac{1}{m}\left(\frac{u}{2}-\frac{\sin 2 u}{4}\right)\right|_{0} ^{m \pi} \\ &=\frac{1}{m}\left[\frac{m \pi}{2}-\frac{\sin 2 m \pi}{4}-0\right]\\ &=\frac{\pi}{2} \end{aligned}
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