$\int \frac{x^2}{\sqrt{4-9 x^2}} \mathit{\ⅆ}x$

$\= \int \frac{{\left(\frac{2}{3}\sin \left(\mathrm{\θ}\right)\right)}^2\left(\frac{2}{3}\cos \left(\mathrm{\θ}\right) d\mathrm{\θ}\right)}{2 \cos \left(\mathrm{\θ}\right)}$

$\= \frac{4}{27}\int {\sin }^2\left(\mathrm{\θ}\right) \mathit{\ⅆ}\mathrm{\θ}$

$\=\frac{4}{27}\int \frac{1}{2}\left(1-\cos \left(2 \mathrm{\θ}\right)\right) \mathit{\ⅆ}\mathrm{\θ}$

$\=\frac{2}{27}\left(\mathrm{\θ}-\frac{\sin \left(2 \mathrm{\θ}\right)}{2}\right)$

$\=\frac{2}{27}\left(\mathrm{\θ}-\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\θ}\right)\right)$

$\=\frac{2}{27}\left(\arcsin \left(\frac{3}{2}x\right)-\left(\frac{3}{2}x\right)\left(\frac{1}{2}\sqrt{4-9 x^2}\right)\right)$

$\=\frac{2}{27} \arcsin \left(\frac{3}{2}x\right)\=\frac{x}{18}\sqrt{4-9 x^2}$

Evaluate the given integral

$\int \frac{x^2}{\sqrt{4-9 x^2}} \mathit{\ⅆ}x$ = $-\frac{1}{18}x\sqrt{-9x^2\+4}\+\frac{2}{27}\arcsin \left(\frac{3}{2}x\right)$$$$$

Initialization

$\mathrm{with}\left(\mathrm{IntegrationTools}\right)\:$

$Q≔\int \frac{x^2}{\sqrt{4-9 x^2}} \mathit{\ⅆ}x\:$

Change variables as per Table 6.3.1

$q_1≔\mathrm{Change}\left(Q\,x\=\frac{2}{3}\sin \left(\mathrm{\θ}\right)\right)$

$\∫\left(-\frac{2}{27}\frac{{\sin \left(\mathrm{\θ}\right)}^2\sqrt{-4{\sin \left(\mathrm{\θ}\right)}^2\+4}\cos \left(\mathrm{\θ}\right)}{{\sin \left(\mathrm{\θ}\right)}^2-1}\right)\ⅆ\mathrm{\θ}$

$q_2≔\mathrm{simplify}\left(q_1\right) assuming \mathrm{\θ}\colon\colon \mathrm{RealRange}\left(-\frac{\mathrm{\π}}{2}\,\frac{\mathrm{\π}}{2}\right)$

$\frac{4}{27}\∫{\sin \left(\mathrm{\θ}\right)}^2\ⅆ\mathrm{\θ}$

$q_3≔\mathrm{simplify}\left(q_2\,\left\{{\sin }^2\left(\mathrm{\θ}\right)\=\left(1- \cos \left(2 \mathrm{\θ}\right)\right)\/2\right\}\right)$

$\frac{4}{27}\∫\left(\frac{1}{2}-\frac{1}{2}\cos \left(2\mathrm{\θ}\right)\right)\ⅆ\mathrm{\θ}$

$q_4≔\mathrm{expand}\left(q_3\,2 \mathrm{\θ}\right)$

$\frac{2}{27}\∫1\ⅆ\mathrm{\θ}-\frac{2}{27}\∫\cos \left(2\mathrm{\θ}\right)\ⅆ\mathrm{\θ}$

$q_5≔\mathrm{value}\left(q_4\right)$

$\frac{2}{27}\mathrm{\θ}-\frac{1}{27}\sin \left(2\mathrm{\θ}\right)$

$q_6≔\mathrm{expand}\left(q_5\right)$

$\frac{2}{27}\mathrm{\θ}-\frac{2}{27}\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\θ}\right)$

$\mathrm{eval}\left(q_6\,\mathrm{\θ}\=\arcsin \left(\frac{3}{2}x\right)\right)$

$-\frac{1}{18}x\sqrt{-9x^2\+4}\+\frac{2}{27}\arcsin \left(\frac{3}{2}x\right)$

$\begin{array}{cccc}\multicolumn{4}{c}{\int \frac{x^2}{\sqrt{-9x^2+4}}\ⅆx}\\ & \text{=}& \frac{4\int {\sin \left(u\right)}^2\ⅆu}{27}\hfill & \left[\mathrm{change}\,x=\frac{2\sin \left(u\right)}{3}\,u\right]\\ & \text{=}& \frac{2u}{27}-\frac{2\int \cos \left(2u\right)\ⅆu}{27}\hfill & \left[\mathrm{rewrite}\,{\sin \left(u\right)}^2=\frac{1}{2}-\frac{\cos \left(2u\right)}{2}\right]\\ & \text{=}& \frac{2u}{27}-\frac{\int \cos \left(\mathrm{u1}\right)\ⅆ\mathrm{u1}}{27}\hfill & \left[\mathrm{change}\,\mathrm{u1}=2u\,\mathrm{u1}\right]\\ & \text{=}& \frac{2u}{27}-\frac{\sin \left(\mathrm{u1}\right)}{27}\hfill & \left[\cos \right]\\ & \text{=}& \frac{2u}{27}-\frac{\sin \left(2u\right)}{27}\hfill & \left[\mathrm{revert}\right]\\ & \text{=}& -\frac{x\sqrt{-9x^2+4}}{18}+\frac{2\arcsin \left(\frac{3x}{2}\right)}{27}\hfill & \left[\mathrm{revert}\right]\end{array}$

Table 6.3.7(a)   Annotated stepwise solution via Integration Methods tutor