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