Using the substitution $u = x^{3/2}$ and then substituting $u = a \sin \theta$ can be a more efficient way to solve this integral. Let's follow this method step-by-step.

Step-by-Step Solution:

1. Substitution $u = x^{3/2}$: $u = x^{3/2} \implies du = \frac{3}{2} x^{1/2} \, dx \implies dx = \frac{2}{3} x^{-1/2} \, du$ Since $x = (u)^{2/3}$, $dx = \frac{2}{3} (u^{2/3})^{-1/2} \, du = \frac{2}{3} u^{-1/3} \, du$

2. Rewrite the integral: $\int \frac{x^{3/2}}{\sqrt{a^2 - x}} \, dx = \int \frac{u}{\sqrt{a^2 - (u)^{2/3}}} \cdot \frac{2}{3} u^{-1/3} \, du = \frac{2}{3} \int \frac{u^{2/3}}{\sqrt{a^2 - u^{2/3}}} \, du$

3. Simplify the integral: $\frac{2}{3} \int \frac{u^{2/3}}{\sqrt{a^2 - u^{2/3}}} \, du$

4. Second substitution $u = a \sin \theta$: $u = a \sin \theta \implies du = a \cos \theta \, d\theta$ The integral becomes: [ \frac{2}{3} \int \frac{(a \sin \theta)^{2/3}}{\sqrt{a^2 - (