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: 45


$$\int 4\tan^3 2xdx=\tan^22x-2\ln|\sec2x|+C$$

Work Step by Step

$$A=\int 4\tan^3 2xdx$$ Use Reduction Formula 93, which states that $$\int\tan^naxdx=\frac{\tan^{n-1}ax}{a(n-1)}-\int\tan^{n-2}axdx$$ for $n=3$ and $a=2$ $$A=4\Big[\frac{\tan^22x}{4}-\int\tan2xdx\Big]$$ Next, use Formula 89, which states that $$\int\tan axdx=\frac{1}{a}\ln|\sec ax|+C$$ for $a=2$ $$A=4\Big[\frac{\tan^22x}{4}-\frac{1}{2}\ln|\sec2x|\Big]+C$$ $$A=\tan^22x-2\ln|\sec2x|+C$$
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