$\int x^2\sqrt{4-9 x^2}\mathit{\ⅆ}x$

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

$\= \frac{16}{27}\int {\sin }^2\left(\mathrm{\θ}\right){\cos }^2\left(\mathrm{\θ}\right) \mathit{\ⅆ}\mathrm{\θ}$

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

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

$\=\frac{2}{27}\left(\mathrm{\θ}-\frac{\sin \left(4 \mathrm{\θ}\right)}{4}\right)$

$\=\frac{2}{27}\left(\mathrm{\θ}-\frac{1}{4}\left(4 \sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\θ}\right)\left(2 {\cos }^2\left(\mathrm{\θ}\right)-1\right)\right)\right)$

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

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

$\=\frac{2}{27} \arcsin \left(\frac{3}{2}x\right)-\frac{x}{36}\left(2-9 x^2\right)\sqrt{4-9 x^2}$

Evaluate the given integral

$\int x^2\sqrt{4-9 x^2}\mathit{\ⅆ}x$ = $-\frac{1}{36}x{\left(-9x^2\+4\right)}^{3\/2}\+\frac{1}{18}x\sqrt{-9x^2\+4}\+\frac{2}{27}\arcsin \left(\frac{3}{2}x\right)$$$$$

Initialization

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

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

Change variables as per Table 6.3.1

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

$\∫\frac{8}{27}{\sin \left(\mathrm{\θ}\right)}^2\sqrt{-4{\sin \left(\mathrm{\θ}\right)}^2\+4}\cos \left(\mathrm{\θ}\right)\ⅆ\mathrm{\θ}$

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

$\frac{16}{27}\∫{\sin \left(\mathrm{\θ}\right)}^2{\cos \left(\mathrm{\θ}\right)}^2\ⅆ\mathrm{\θ}$

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

$\frac{16}{27}\∫\frac{1}{4}{\sin \left(2\mathrm{\θ}\right)}^2\ⅆ\mathrm{\θ}$

$q_4≔\mathrm{eval}\left(q_3\,{\sin }^2\left(2 \mathrm{\θ}\right)\=\left(1-\cos \left(4 \mathrm{\θ}\right)\right)\/2\right)$

$\frac{16}{27}\∫\left(\frac{1}{8}-\frac{1}{8}\cos \left(4\mathrm{\θ}\right)\right)\ⅆ\mathrm{\θ}$

$q_5≔\mathrm{expand}\left(q_4\,4 \mathrm{\θ}\right)$

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

$q_6≔\mathrm{value}\left(q_5\right)$

$\frac{2}{27}\mathrm{\θ}-\frac{1}{54}\sin \left(4\mathrm{\θ}\right)$

$q_7≔\mathrm{expand}\left(q_6\right)$

$\frac{2}{27}\mathrm{\θ}-\frac{4}{27}\sin \left(\mathrm{\θ}\right){\cos \left(\mathrm{\θ}\right)}^3\+\frac{2}{27}\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\θ}\right)$

$\mathrm{eval}\left(q_7\,\mathrm{\θ}\=\arcsin \left(\frac{3}{2}x\right)\right)$

$-\frac{1}{36}x{\left(-9x^2\+4\right)}^{3\/2}\+\frac{1}{18}x\sqrt{-9x^2\+4}\+\frac{2}{27}\arcsin \left(\frac{3}{2}x\right)$

$\begin{array}{c}\int x^2\sqrt{-9x^2+4}\ⅆx\hfill \\ & \text{=}& \frac{16\int {\sin \left(u\right)}^2{\cos \left(u\right)}^2\ⅆu}{27}\hfill & \left[\mathrm{change}\,x=\frac{2\sin \left(u\right)}{3}\,u\right]\\ & \text{=}& -\frac{4\int \left(1+\cos \left(2u\right)\right)\left(-1+\cos \left(2u\right)\right)\ⅆu}{27}\hfill & \left[\mathrm{rewrite}\,{\cos \left(u\right)}^2=\frac{\cos \left(2u\right)}{2}+\frac{1}{2}\,{\sin \left(u\right)}^2=\frac{1}{2}-\frac{\cos \left(2u\right)}{2}\right]\\ & \text{=}& -\frac{4\int {\cos \left(2u\right)}^2\ⅆu}{27}+\frac{4u}{27}\hfill & \left[\mathrm{rewrite}\,\left(1+\cos \left(2u\right)\right)\left(-1+\cos \left(2u\right)\right)={\cos \left(2u\right)}^2-1\right]\\ & \text{=}& \frac{2u}{27}-\frac{2\int \cos \left(4u\right)\ⅆu}{27}\hfill & \left[\mathrm{rewrite}\,{\cos \left(2u\right)}^2=\frac{1}{2}+\frac{\cos \left(4u\right)}{2}\right]\end{array}$

Table 6.3.6(a)   First steps of an annotated stepwise solution via Integration Methods tutor