University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 8 - Section 8.5 - Integral Tables and Computer Algebra Systems - Exercises - Page 451: 26


$$\int\cos\frac{\theta}{2}\cos7\theta d\theta=\frac{\sin\frac{13\theta}{2}}{13}+\frac{\sin\frac{15\theta}{2}}{15}+C$$

Work Step by Step

$$A=\int\cos\frac{\theta}{2}\cos7\theta d\theta$$ Use Formula 69c, which states that $$\int \cos ax\cos bx dx=\frac{\sin(a-b)x}{2(a-b)}+\frac{\sin(a+b)x}{2(a+b)}+C$$ for $a=1/2$ and $b=7$: $a-b=1/2-7=-13/2$ while $a+b=1/2+7=15/2$ $$A=\frac{\sin\Big(-\frac{13}{2}\Big)\theta}{2\times\Big(-\frac{13}{2}\Big)}+\frac{\sin\frac{15}{2}\theta}{2\times\frac{15}{2}}+C$$ We have $\sin\Big(-\frac{13}{2}\Big)\theta=-\sin\frac{13}{2}\theta$ $$A=\frac{-\sin\frac{13\theta}{2}}{-13}+\frac{\sin\frac{15\theta}{2}}{15}+C$$ $$A=\frac{\sin\frac{13\theta}{2}}{13}+\frac{\sin\frac{15\theta}{2}}{15}+C$$
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