Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.3 Trigonometric Substitution - Exercises - Page 410: 26


$$\ln \left|\frac{y}{3}+\frac{\sqrt{y^{2}-9}}{3}\right|+C$$

Work Step by Step

Given $$\int \frac{d y}{\sqrt{y^{2}-9}}$$ Let $$y=3\sec u\ \ \ \ \ \ \to \ \ \ \ \ dy=3\sec u\tan udu $$ Then \begin{align*} \int \frac{d y}{\sqrt{y^{2}-9}}&=\int \frac{3\sec u\tan udu}{\sqrt{9\sec^2u-9}}\\ &=\int \frac{3\sec u\tan udu}{\sqrt{9\tan^2u}}\\ &=\int\sec udu\\ &=\ln|\sec u+\tan u|+C\\ &=\ln \left|\frac{y}{3}+\frac{\sqrt{y^{2}-9}}{3}\right|+C \end{align*}
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