Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - 7.3 Trigonometric Substitution - 7.3 Exercises - Page 531: 1



Work Step by Step

Given $$\int\frac{dx}{x^2\sqrt{4-x^2}} $$ Let $x=2\sin \theta \ \Rightarrow \ dx=2\cos \theta d \theta$, then \begin{align*} \int\frac{dx}{x^2\sqrt{4-x^2}}&=\int\frac{2\cos \theta d \theta}{4\sin^2 \theta\sqrt{4-4\sin^2 \theta}}\\ &=\int\frac{2\cos \theta d \theta}{4\sin^2 \theta(2\cos \theta)}\\ &=\frac{1}{4}\int \csc^2\theta d\theta\\ &=\frac{-1}{4}\cot \theta +c\\ &=\frac{-\sqrt{4-x^2}}{4x}+c \end{align*}
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