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: 16


$$\ln|\cosh x|-\frac{1}{2}\tanh ^2x+C$$

Work Step by Step

\begin{aligned} \int \tanh ^{3} x d x &=\int \tanh ^{2} x \tanh x d x \\ &=\int\left(1-\operatorname{sech}^{2} x\right) \tanh x d x \\ &=\int \tanh x-\operatorname{sech}^{2} x \tanh x d x \\ &=\int \tanh x d x-\int \operatorname{sech}^{2} x \tanh x d x \\ &=\ln|\cosh x|-\frac{1}{2}\tanh ^2x+C \end{aligned}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.