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: 19


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

Work Step by Step

Given $$\int \frac{d x}{\sqrt{x^{2}-9}} $$ Let $$x=3\sec \theta ,\ \ \ dx= 3\sec \theta \tan \theta d \theta $$ Then \begin{aligned} \int \frac{d x}{\sqrt{x^{2}-9}} &=\int \frac{3 \sec \theta \tan \theta d \theta}{\sqrt{(3 \sec \theta)^{2}-9}} \\ &=\int \frac{3 \sec \theta \tan \theta d \theta}{\sqrt{9 \sec ^{2} \theta-9}} \\ &=\int \frac{3 \sec \theta \tan \theta d \theta}{3 \sqrt{\sec ^{2} \theta-1}} \\ &=\int \frac{\sec \theta \tan \theta d \theta}{\sqrt{\tan ^{2} \theta}} \\ &=\int \sec \theta d \theta \\ &=\ln |\sec \theta+\tan \theta|+C\\ &=\ln \left|\frac{x+\sqrt{x^{2}-9}}{3}\right|+C \end{aligned}
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