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: 21


$$\frac{1}{2}\left[x \sqrt{x^{2}-1}-\cosh ^{-1} x\right]+C$$

Work Step by Step

Given $$\int \sqrt{x^{2}-1} d x $$ Let $$x=\cosh t\ \ \to \ dx=\sinh udu$$ \begin{aligned} \int \sqrt{x^{2}-1} d x &=\int \sqrt{\cosh ^{2} t-1} \sinh t d t \\ &=\int \sqrt{\sinh ^{2} t} \sinh t d t \\ &=\int \sinh t \cdot \sinh t d t \\ &=\int \sinh ^{2} t d t \\ &=\frac{1}{2}\left[\int \cosh 2 t d t-\int 1 d t\right] \\ &=\frac{1}{2}\left[\frac{1}{2} \sinh 2 t-t\right]+C\\ &=\frac{1}{2}\sinh t\cosh t-\frac{1}{2}t+C\\ &=\frac{1}{2}\left[x \sqrt{x^{2}-1}-\cosh ^{-1} x\right]+C \end{aligned}
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