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: 9


$$-\frac{x}{4 \sqrt{x^{2}-4}}+C$$

Work Step by Step

Given$$\int \frac{d x}{\left(x^{2}-4\right)^{\frac{3}{2}}} $$ Let, $x=2 \sec \theta, \quad d x=2 \sec \theta \tan \theta d \theta$ \begin{aligned} \int \frac{d x}{\left(x^{2}-4\right)^{\frac{3}{2}}} &=\int \frac{2 \sec \theta \tan \theta d \theta}{\left.(2 \sec \theta)^{2}-4\right)^{\frac{3}{2}}} \\ &=\int \frac{2 \sec \theta \tan \theta d \theta}{\left(4 \sec ^{2} \theta-4\right)^{\frac{3}{2}}} \\ &=\int \frac{2 \sec \theta \tan \theta d \theta}{4^{\frac{3}{2}}\left(\sec ^{2} \theta-1\right)^{\frac{3}{2}}} \\ &=\int \frac{2 \sec \theta \tan \theta d \theta}{8 \tan ^{3} \theta} \\ &=\int \frac{2 \sec \theta d \theta}{8 \tan ^{2} \theta}\\ &=\frac{1}{4}\int \sin^{-2}\theta \cos \theta d\theta\\ &=\frac{-1}{4}\frac{1}{\sin \theta} +C\\ &=\frac{-1}{4}\frac{1}{\sqrt{1-\cos^2\theta}} +C\\ &=-\frac{x}{4 \sqrt{x^{2}-4}}+C \end{aligned}
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