The substitution $x\=\frac{2}{3}\tan \left(\mathrm{\θ}\right)$ means $\mathrm{dx}\=\frac{2}{3}{\sec }^2\left(\mathrm{\θ}\right) d\mathrm{\θ}$, and turns $g\left(x\right)$ into $2 \sec \left(\mathrm{\θ}\right)$. From Figure 6.3.2, $\sec \left(\mathrm{\θ}\right)\=\frac{1}{2}\sqrt{4\+9 x^2}$. Hence, the evaluation of the given integral proceeds as follows.

The integral in the second line is evaluated by writing it in the form $\int u^2 \mathrm{dx}$ by setting $u\=\sec \left(\mathrm{\θ}\right)$. However, there is an "easier" solution to the original integral, obtained by the substitution $u\=4\+9 x^2$ so that $\mathrm{du}\=18 x \mathrm{dx}$. With this substitution, the integral becomes $\int u^{1\/2} \frac{\mathit{\ⅆ}u}{18}$ whose value is

$\frac{u^{3\/2}}{18\cdot \left(3\/2\right)}\=\frac{u^{3\/2}}{27}\=\frac{{\left(4\+9 x^2\right)}^{3\/2}}{27}$ 

in agreement with the solution obtained by the trig substitution.

A stepwise solution that uses top-level commands except for one application of the Change command from the IntegrationTools package:

The stepwise solution provided by the  tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution $u^2\=9 x^2\+4$ that results immediately in the integral $\frac{1}{9}\int u^2 \mathrm{du}$.

On the other hand, Table 6.3.11(a) shows the result when the Change rule $x\=\frac{2}{3}\tan \left(\mathrm{\θ}\right)$ is imposed on the tutor. The integrand is written as $\tan \left(\mathrm{\θ}\right){\sec }^3\left(\mathrm{\θ}\right)$ and its antiderivative is found by the Change rule with $u\=\sec \left(\mathrm{\θ}\right)$.