$\int^\alpha_0 \sqrt\frac{x^3}{a^3 – x^3} dx$ $= \frac{a\sqrt\pi \sqrt{5/6}}{\sqrt{1/3}}$

Solution:

Let $I=\int_{0}^{a} \sqrt{\frac{x^{3}}{a^{3}-x^{3}}} d x$

Put $x^{3}=a^{3} t \quad \Rightarrow \quad x=a t^{\frac{1}{3}}$

Diff. w.r.t. x

$d x=\frac{a}{3} t^{-2 / 3}$ dt

$\begin{aligned} \text { Limits becomes } \rightarrow[0,1] & \\ I =\int_{0}^{1}(t)^{3 / 2} \cdot(1-t)^{3 / 2} \cdot t^{-2 / 3} \frac{a}{3} d t \\ =\frac{a}{3} \int_{0}^{1} t^{5 / 6}(1-t)^{3 / 2} d t \\=\frac{a}{3} \beta\left(\frac{5}{6}, \frac{3}{2}\right) \end{aligned}$

$I=a \frac{\sqrt{\pi} \Gamma\left(\frac{5}{6}\right)}{\Gamma\left(\frac{1}{3}\right)}$