Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.9 Hyperbolic Functions - Exercises - Page 385: 64


$$\int \operatorname{sech} x \, d x=\tan ^{-1}(\sinh x)+C$$

Work Step by Step

Given $$\int \operatorname{sech} x \, d x $$ Let $u=\sinh x,\ \ du=\cosh xdx $. Then \begin{align*} \int \operatorname{sech} x \, d x &= \int \frac{1}{\cosh x}dx\\ &= \int \frac{1}{\cosh^2 x}du\\ &=\int \frac{1}{1+\sinh ^{2} u} d u\\ &=\int \frac{1}{1+u^{2}} du\\ &=\tan ^{-1}u+C\\ &=\tan ^{-1}(\sinh x)+C \end{align*}
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