Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 460: 14


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

Work Step by Step

Given $$\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x$$ Let $$ x=3\sin u\ \ \ \ \ \ \ \ dx=3\cos udu $$ Then \begin{align*} \int \frac{x^{2}}{\sqrt{9-x^{2}}} d x&=\int \frac{9\sin^{2}u}{\sqrt{9-9\sin^{2}u}} 3\cos udu\\ &= \int \frac{9\sin^{2}u}{3\cos u} 3\cos udu\\ &= 9\int \sin^2u du\\ &=\frac{9}{2}\int (1-\cos 2u)du\\ &= \frac{9}{2}\left( u-\frac{1}{2}\sin 2u\right)+C\\ &= \frac{9}{2}\left( u- \sin u\cos u\right)+C\\ &= \frac{9}{2}\left( \sin ^{-1}\left(\frac{x}{3}\right)- \frac{x}{3} \frac{\sqrt{9-x^{2}}}{3}\right)+C \end{align*}
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