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 8.1 - Using Basic Integration Formulas - Exercises 8.1 - Page 448: 25


$$\int \frac{d y}{ \sqrt{e ^{2y}-1}}=\sec ^{-1}\left|e^{y}\right|+C $$

Work Step by Step

Given $$\int \frac{d y}{ \sqrt{e ^{2y}-1}}$$ Let $$ u=e^y \Rightarrow y=\ln u\Rightarrow d y=\frac{d u}{u}$$ So, we have \begin{aligned} I&= \int \frac{d u}{u \sqrt{u^{2}-1}}\\ &=\sec ^{-1}|u|+C\\ &=\sec ^{-1}\left|e^{y}\right|+C \end{aligned}
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