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: 15


$$\frac{9}{2} \sin ^{-1} \frac{x}{3}-\frac{1}{2} x \sqrt{9-x^{2}}+C$$

Work Step by Step

Given $$\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x $$ Let $x=3 \sin \theta, \quad d x=3 \cos \theta d \theta$ \begin{aligned} \int \frac{x^{2}}{\sqrt{9-x^{2}}} d x &=\int \frac{9 \sin ^{2} \theta}{\sqrt{9-9 \sin ^{2} \theta}} \cdot 3 \cos \theta d \theta \\ &=\int \frac{9 \sin ^{2} \theta}{3 \sqrt{1-\sin ^{2} \theta}} \cdot 3 \cos \theta d \theta \\ &=\int \frac{9 \sin ^{2} \theta}{\sqrt{\cos ^{2} \theta}} \cdot \cos \theta d \theta \\ &=9 \int \frac{\sin ^{2} \theta}{\cos \theta} \cdot \operatorname{ecs} \theta d \theta \\ &=9 \int \sin ^{2} \theta d \theta\\ &=\frac{9}{2}\int (1-\cos 2\theta )d\theta\\ &=\frac{9}{2}\left(\theta -\frac{1}{2}\sin 2\theta \right)+C\\ &=\frac{9}{2}\left(\theta -\sin \theta\cos \theta \right)+C\\ &=\frac{9}{2} \sin ^{-1} \frac{x}{3}-\frac{1}{2} x \sqrt{9-x^{2}}+C \end{aligned}
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