Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 42


$$2 \tan ^{-1}\left(\tanh \frac{x}{2}\right)+C$$

Work Step by Step

Given $$\int \operatorname{sech} x \, d x$$ Let $u=\tanh (x/2)$ $$ \cosh x=\frac{1+u^{2}}{1-u^{2}}, \quad \sinh x=\frac{2 u}{1-u^{2}}, \quad d x=\frac{2 d u}{1-u^{2}}$$ Then \begin{aligned} \int \operatorname{sech} x \, d x &=\int \frac{1-u^{2}}{1+u^{2}}\left(\frac{2 d u}{1-u^{2}}\right) \\ &=2 \int \frac{d u}{1+u^{2}} \\ &=2\left(\tan ^{-1} u+C\right)\\ &= 2 \tan ^{-1}\left(\tanh \frac{x}{2}\right)+C \end{aligned}
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