Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Section 8.6 - Integral Tables and Computer Algebra Systems - Exercises 8.6 - Page 481: 47

Answer

$\dfrac{\sec\pi x\tan\pi x}{\pi}+\dfrac{1}{\pi}\ln|\sec \pi x+\tan\pi x|+C$

Work Step by Step

Apply the Reduction Formula: $\int\sec^naxdx=\dfrac{\sec^{(n-2)}ax\tan ax}{a(n-1)}+\dfrac{n-2}{n-1}\int\sec^{n-2}axdx$ Let $I=2[\dfrac{\sec\pi x\tan \pi x}{2\pi}+\dfrac{1}{2}\int\sec\pi xdx] \\=\dfrac{\sec\pi x\tan\pi x}{\pi}+\int\sec\pi xdx \\=\dfrac{\sec\pi x\tan\pi x}{\pi}+\dfrac{1}{\pi}\ln|\sec \pi x+\tan\pi x|+C$
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