Calculus 8th Edition

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

Chapter 7 - Techniques of Integration - 7.5 Strategy for Integration - 7.5 Exercises - Page 548: 38

Answer

$$\displaystyle{\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\frac{\sin \theta \cot \theta}{\sec \theta}d\theta}=\frac{\pi}{12}}$$

Work Step by Step

$\displaystyle{I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\frac{\sin \theta \cot \theta}{\sec \theta}d\theta}\\ I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\sin \theta \times \frac{\cos \theta}{\sin \theta}\times \cos \theta \ d\theta}\\ I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\cos ^2\theta \,\,d\theta}\\ I=\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\cos 2\theta +1\,\,d\theta}\\ I=\frac{1}{2}\left[ \frac{1}{2}\sin 2\theta +\theta \right] _{\frac{\pi}{6}}^{\frac{\pi}{3}}\\ I=\frac{\pi}{12}}$
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