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