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: 21


$$-\frac{\sqrt{5-y^{2}}}{5 y}+C$$

Work Step by Step

Given $$ \int \frac{d y}{y^{2} \sqrt{5-y^{2}}} $$ Let $$ y= \sqrt{5} \sin \theta ,\ \ \ dy= \sqrt{5}\cos \theta d \theta $$ Then \begin{aligned} \int \frac{d y}{y^{2} \sqrt{5-y^{2}}} &=\int \frac{\sqrt{5} \cos \theta d \theta}{(\sqrt{5} \sin \theta)^{2} \sqrt{5-(\sqrt{5} \sin \theta)^{2}}} \\ &=\int \frac{\sqrt{5} \cos \theta d \theta}{5 \sin ^{2} \theta \sqrt{5-5 \sin ^{2} \theta}} \\ &=\int \frac{\sqrt{5} \cos \theta d \theta}{5 \sqrt{5} \sin ^{2} \theta \sqrt{1-\sin ^{2} \theta}} \\ &=\frac{1}{5} \int \frac{\cos \theta d \theta}{\sin ^{2} \theta \sqrt{\cos ^{2} \theta}} \\ &=\frac{1}{5} \int \frac{\cos \theta d \theta}{\sin ^{2} \theta \cos \theta}\\ &=\frac{1}{5} \int \csc ^{2} \theta d \theta\\ &=\frac{1}{5}(-\cot \theta)+C\\ &= -\frac{\sqrt{5-y^{2}}}{5 y}+C \end{aligned}
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