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: 37


$$\frac{1}{2} \sec ^{-1}\left(\frac{x}{2}\right)+C$$

Work Step by Step

Given $$\int \frac{d x}{x \sqrt{x^{2}-4}}$$ Let $$x=2\sec u \ \ \ \ \ \ \ \ dx=2\sec u\tan udu $$ Then \begin{align*} \int \frac{d x}{x \sqrt{x^{2}-4}}&=\int \frac{2\sec u\tan udu}{2\sec u \sqrt{4\sec^2 u-4}}\\ &= \int \frac{2\sec u\tan udu}{2\sec u \sqrt{4\tan^2 u}}\\ &=\frac{1}{2}\int du\\ &=\frac{1}{2}u+C\\ &= \frac{1}{2} \sec ^{-1}\left(\frac{x}{2}\right)+C \end{align*}
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