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

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

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

$\=3\int \frac{1-{\sin }^2\left(\mathrm{\θ}\right)}{{\sin }^2\left(\mathrm{\θ}\right)} \mathit{\ⅆ}\mathrm{\θ}$

$\=3\int \left({\csc }^2\left(\mathrm{\θ}\right)-1\right) \mathit{\ⅆ}\mathrm{\θ}$

$\= -3\left(\cot \left(\mathrm{\θ}\right)\+\mathrm{\θ}\right)$

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

Evaluate the given integral

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

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

$\stackrel{\text{simplify}}{\=}$

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

$\stackrel{\text{expand}}{\=}$

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

Initialization

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

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

Change variables as per Table 6.3.1

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

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

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

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

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

$3\left(\∫\frac{1-{\sin \left(\mathrm{\θ}\right)}^2}{{\sin \left(\mathrm{\θ}\right)}^2}\ⅆ\mathrm{\θ}\right)$

$q_4≔\mathrm{expand}\left(q_3\right)$

$-3\left(\∫\left(1-\frac{1}{{\sin \left(\mathrm{\θ}\right)}^2}\right)\ⅆ\mathrm{\θ}\right)$

$q_5≔\mathrm{trigsubs}\left(\sin \left(\mathrm{\θ}\right)\=1\/ \csc \left(\mathrm{\θ}\right)\,q_4\right)$

$-3\left(\∫\left(1-{\csc \left(\mathrm{\θ}\right)}^2\right)\ⅆ\mathrm{\θ}\right)$

$q_6≔\mathrm{expand}\left(q_5\right)$

$-3\left(\∫1\ⅆ\mathrm{\θ}\right)\+3\left(\∫{\csc \left(\mathrm{\θ}\right)}^2\ⅆ\mathrm{\θ}\right)$

$q_7≔\mathrm{value}\left(q_6\right)$

$-3\mathrm{\θ}-\frac{3\cos \left(\mathrm{\θ}\right)}{\sin \left(\mathrm{\θ}\right)}$

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

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

$\begin{array}{cccc}\multicolumn{4}{c}{\int \frac{\sqrt{-9x^2+4}}{x^2}\ⅆx}\\ & \text{=}& 3\int {\cot \left(\mathrm{\theta }\right)}^2\ⅆ\mathrm{\theta }\hfill & \left[\mathrm{change}\,x=\frac{2\sin \left(\mathrm{\theta }\right)}{3}\right]\\ & \text{=}& 3\int {\csc \left(\mathrm{\theta }\right)}^2\ⅆ\mathrm{\theta }-3\mathrm{\theta }\hfill & \left[\mathrm{rewrite}\,{\cot \left(\mathrm{\theta }\right)}^2={\csc \left(\mathrm{\theta }\right)}^2-1\right]\end{array}$

Table 6.3.8(a)   First steps of an annotated stepwise solution via Integration Methods tutor