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

$\= \int \frac{\left(\frac{2}{3}\sec \left(\mathrm{\θ}\right)\right)\left(\frac{2}{3}\sec \left(\mathrm{\θ}\right)\tan \left(\mathrm{\θ}\right) d\mathrm{\θ}\right)}{2 \tan \left(\mathrm{\θ}\right)}$

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

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

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

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

Evaluate the given integral

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

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

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

Change variables as per Table 6.3.1

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

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

$q_2≔\mathrm{simplify}\left(q_1\right) assuming \mathrm{\θ}\colon\colon \mathrm{RealRange}\left(0\,\frac{\mathrm{\π}}{2}\right)$

$\frac{2}{9}\∫\frac{1}{{\cos \left(\mathrm{\θ}\right)}^2}\ⅆ\mathrm{\θ}$

$q_3≔\mathrm{value}\left(q_2\right)$

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

$\mathrm{simplify}\left(\mathrm{eval}\left(q_3\,\mathrm{\θ}\=\mathrm{arcsec}\left(\frac{3}{2}x\right)\right)\right) assuming x\colon\colon \mathrm{RealRange}\left(0\,\mathrm{\π}\/2\right)$

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

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

Table 6.3.20(a)   The substitution $u^2\=9 x^2-4$ made by the Integration Methods tutor

$\begin{array}{cccc}\multicolumn{4}{c}{\int \frac{x}{\sqrt{9x^2-4}}\ⅆx}\\ & \text{=}& \frac{2\int \sec \left(\mathrm{\theta }\right)\left|\sec \left(\mathrm{\theta }\right)\right|\ⅆ\mathrm{\theta }}{9}\hfill & \left[\mathrm{change}\,x=\frac{2\sec \left(\mathrm{\theta }\right)}{3}\right]\\ & \text{=}& \frac{2\int {\sec \left(\mathrm{\theta }\right)}^2\ⅆ\mathrm{\theta }}{9}\hfill & \left[\mathrm{rewrite}\,\left|\sec \left(\mathrm{\theta }\right)\right|=\sec \left(\mathrm{\theta }\right)\right]\\ & \text{=}& \frac{2u}{9}\hfill & \left[\mathrm{change}\,u=\tan \left(\mathrm{\theta }\right)\,u\right]\\ & \text{=}& \frac{2\tan \left(\mathrm{\theta }\right)}{9}\hfill & \left[\mathrm{revert}\right]\\ & \text{=}& \frac{\sqrt{9x^2-4}}{9}\hfill & \left[\mathrm{revert}\right]\end{array}$

Table 6.3.20(b)   Integration Methods tutor after $x\=\frac{2}{3}\sec \left(\mathrm{\theta }\right)$ is imposed