Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 460: 65


$$\tan^{-1}\left(\tanh ^{-1} t\right)+C$$

Work Step by Step

Given $$\int \frac{d t}{\cosh ^{2} t+\sinh ^{2} t}$$ Let $$ u=\tanh t\ \ \ \ \ \ du=\operatorname{sech}^{2} t d t $$ Then \begin{align*} \int \frac{d t}{\cosh ^{2} t+\sinh ^{2} t}&=\int \frac{d t}{\cosh ^{2} t(1+\tanh ^{2}t) }\\ &=\int \frac{\operatorname{sech}^{2} t d t}{1+\tanh ^{2} t}\\ &=\int \frac{d u}{1+u^{2}}\\ &=\tan^{-1} u+C\\ &=\tan^{-1}\left(\tanh ^{-1} t\right)+C \end{align*}
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