Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - Review - Exercises - Page 578: 11


$$\sqrt{3} -\frac{\pi }{3}$$

Work Step by Step

Given $$\int_{1}^{2} \frac{\sqrt{x^{2}-1}}{x} d x $$ Let $u^2=x^2-1\ \Rightarrow 2udu=2xdx $, at $x=1\to u=0$, $x=2\to u=\sqrt{3}$, then \begin{align*} \int_{1}^{2} \frac{\sqrt{x^{2}-1}}{x} d x&=\int_{0}^{\sqrt{3}} \frac{u^2du}{u^2+1} \\ &=\int_{0}^{\sqrt{3}} \left(1-\frac{1}{u^2+1} \right)du\\ &=u-\tan^{-1}u\bigg|_{0}^{\sqrt{3}}\\ &=\sqrt{3} -\frac{\pi }{3} \end{align*}
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