Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 57


$$\ln \left|\frac{x}{6}+\frac{\sqrt{x^{2}-36}}{6}\right|+C$$

Work Step by Step

Given $$\int \frac{d x}{\sqrt{x^{2}-36}}$$ Let $$ x=6\sec u \ \ \ \ \ \ dx=6\sec u\tan udu $$ Then \begin{aligned} \int \frac{d x}{\sqrt{x^{2}-36}}&=\int \frac{1}{\sqrt{36 \sec ^{2} u-36}} 6 \sec u \tan u \, d u\\ &=\int \frac{1}{6 \sqrt{\sec ^{2} u-1}} 6 \sec u \tan u \, d u\\ &=\int \frac{1}{\tan u} \sec u \tan u \, d u\\ &=\int \sec u \, d u\\ &=\ln|\sec u+\tan u|+C\\ &= \ln \left|\frac{x}{6}+\frac{\sqrt{x^{2}-36}}{6}\right|+C \end{aligned}
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