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: 30


$$\frac{1}{2}\left[-\operatorname{csch}^{-1}(81)+\operatorname{csch}^{-1}(1)\right] $$

Work Step by Step

\begin{aligned} \int_{1}^{9} \frac{1}{x \sqrt{x^{4}+1}} d x &=\int_{1}^{9} \frac{1}{2} \cdot \frac{2 x}{x^{2} \sqrt{\left(x^{2}\right)^{2}+1}} d x \\ &=\frac{1}{2}\left[-\operatorname{csch}^{-1}\left(x^{2}\right)\right]_{1}^{9} \\ &=\frac{1}{2}\left[-\operatorname{csch}^{-1}(81)+\operatorname{csch}^{-1}(1)\right] \end{aligned}
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