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: 62



Work Step by Step

Given $$ \int_{0}^{\pi/4}\sin 7 x\cos 2xdx$$ Use $$ \sin (a x) \cos (b x)=\frac{1}{2} \sin ((a-b) x)+\frac{1}{2} \sin ((a+b) x) $$ Then \begin{align*} \int_{0}^{\pi/4}\sin 7 x\cos 2xdx&=\frac{1}{2}\int_{0}^{\pi/4}(\sin (5x)+ \sin 9x)dx\\ &=\frac{ -1}{2}\left(\frac{1}{5}\cos5 x+\frac{1}{9}\cos9 x\right)\bigg|_{0}^{\pi/4} \\ &=\frac{\sqrt{2}+7}{45} \end{align*}
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