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

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

$\= \frac{2}{9}\int \sin \left(\mathrm{\θ}\right) \mathit{\ⅆ}\mathrm{\θ}$

$\= -\frac{2}{9}\cos \left(\mathrm{\θ}\right)$

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

$\= -\frac{1}{9}\sqrt{4-9 x^2}$

Evaluate the given integral

$$$\int \frac{x}{\sqrt{4-9 x^2}} \mathit{\ⅆ}x$ = $\frac{1}{9}\frac{\left(3x-2\right)\left(3x\+2\right)}{\sqrt{-9x^2\+4}}$$\stackrel{\text{simplify}}{\=}$$-\frac{1}{9}\sqrt{-9x^2\+4}$$$

Initialization

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

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

Change variables as per Table 6.3.1

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

$\∫\left(-\frac{1}{9}\frac{\sqrt{-4{\sin \left(\mathrm{\θ}\right)}^2\+4}\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\θ}\right)}{{\sin \left(\mathrm{\θ}\right)}^2-1}\right)\ⅆ\mathrm{\θ}$

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

$\frac{2}{9}\∫\sin \left(\mathrm{\θ}\right)\ⅆ\mathrm{\θ}$

$q_3≔\mathrm{value}\left(q_2\right)$

$-\frac{2}{9}\cos \left(\mathrm{\θ}\right)$

$\mathrm{eval}\left(q_3\,\mathrm{\θ}\=\arcsin \left(\frac{3}{2}x\right)\right)$

$-\frac{1}{9}\sqrt{-9x^2\+4}$

$\begin{array}{cccc}\multicolumn{4}{c}{\int \frac{x}{\sqrt{-9x^2+4}}\ⅆx}\\ & \text{=}& \int -\frac{1}{9}\ⅆu\hfill & \left[\mathrm{change}\,-9x^2+4=u^2\,u\right]\\ & \text{=}& -\frac{u}{9}\hfill & \left[\mathrm{constant}\right]\\ & \text{=}& -\frac{\sqrt{-9x^2+4}}{9}\hfill & \left[\mathrm{revert}\right]\end{array}$

Table 6.3.4(a)   Annotated stepwise solution via Integration Methods tutor