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: 11


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

Work Step by Step

Given $$\int \frac{d x}{x^{2} \sqrt{4-x^{2}}}$$ Let $$x=2\sin u\ \ \ \ \ \ \ dx= 2\cos udu $$ Then \begin{align*} \int \frac{d x}{x^{2} \sqrt{4-x^{2}}}&=\int \frac{2\cos udu}{4\sin^{2}u \sqrt{4-4\sin^{2}u}}\\ &=\int \frac{2\cos udu}{4\sin^{2}u \sqrt{4cos^{2}u}}\\ &=\frac{1}{4}\int \csc^2 udu\\ &=\frac{-1}{4}\cot u+C\\ &=\frac{-\sqrt{4-x^{2}}}{4 x}+C \end{align*}
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