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: 42


$$\int 8\cos^42\pi tdt=\frac{\cos^32\pi t\sin2\pi t}{\pi}+\frac{3\sin4\pi t}{4\pi}+3t+C$$

Work Step by Step

$$A=\int 8\cos^42\pi tdt$$ Use Reduction Formula 68, which states that $$\int \cos^nax=\frac{\cos^{n-1}ax\sin ax}{na}+\frac{n-1}{n}\int\cos^{n-2}axdx$$ for $a=2\pi$ and $n=4$ $$A=8\Big[\frac{\cos^32\pi t\sin2\pi t}{8\pi}+\frac{3}{4}\int\cos^22\pi tdt\Big]$$ Use Reduction Formula 68 one more time, this time for $a=2\pi$ and $n=2$ $$A=8\Big[\frac{\cos^32\pi t\sin2\pi t}{8\pi}+\frac{3}{4}\Big(\frac{\cos2\pi t\sin2\pi t}{4\pi}+\frac{1}{2}\int\cos^02\pi tdt\Big)\Big]$$ $$A=8\Big[\frac{\cos^32\pi t\sin2\pi t}{8\pi}+\frac{3}{4}\Big(\frac{\frac{1}{2}\sin4\pi t}{4\pi}+\frac{1}{2}\int 1dt\Big)\Big]$$ $$A=8\Big[\frac{\cos^32\pi t\sin2\pi t}{8\pi}+\frac{3}{4}\Big(\frac{\sin4\pi t}{8\pi}+\frac{t}{2}\Big)\Big]+C$$ $$A=\frac{\cos^32\pi t\sin2\pi t}{\pi}+6\Big(\frac{\sin4\pi t}{8\pi}+\frac{t}{2}\Big)+C$$ $$A=\frac{\cos^32\pi t\sin2\pi t}{\pi}+\frac{3\sin4\pi t}{4\pi}+3t+C$$
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