Answer
$$
\int \frac{x^{2}}{\left(4-x^{2}\right)^{3 / 2}} d x= \frac{x}{\sqrt{4-x^{2}}}-\sin ^{-1}\left(\frac{x}{2}\right)+C
$$
$C$ is constant.
Work Step by Step
$$
\int \frac{x^{2}}{\left(4-x^{2}\right)^{3 / 2}} d x
$$
$$
\text { Let } x=2 \sin \theta \Rightarrow\left(4-x^{2}\right)^{3 / 2}=(2 \cos \theta)^{3}, d x=2 \cos \theta d \theta,
$$
so
$$ \begin{aligned} \int \frac{x^{2}}{\left(4-x^{2}\right)^{3 / 2}} d x & =\int \frac{4 \sin ^{2} \theta}{8 \cos ^{3} \theta} 2 \cos \theta d \theta
\\
&=\int \tan ^{2} \theta d \theta\\
&= \int (\sec ^{2} \theta -1) d \theta \\
&= \tan \theta-\theta+C \\
&= \frac{x}{\sqrt{4-x^{2}}}-\sin ^{-1}\left(\frac{x}{2}\right)+C
\end{aligned} $$
$C$ is constant.
