Showing $\frac{1}{\sqrt{x(1-x)}}$ is integrable on $(0,1)$.

Question

Question: Show that the function $f(x) = \frac{1}{\sqrt{x(1-x)}}$ is Lebesgue-integrable on $(0,1)$.

Any help would be appreciated, thanks.

Answer 1

It can be bounded above by a multiple of $$\frac1{\sqrt x}+\frac1{\sqrt{1-x}}$$ which is $L^1$ on $(0,1)$.

Answer 2

Just another route. One may observe that, by the chain rule, $$ \left(\arcsin \left(\sqrt{1-x} \right)\frac{}{}\right)'=\color{red}{-\frac1{2\sqrt{1-x}}}\cdot \frac1{\sqrt{1-(1-x)}}=-\frac1{2\sqrt{x(1-x)}} $$ giving $$ \int_0^1 \frac1{\sqrt{x(1-x)}}\,dx=\left[-2\arcsin \left(\sqrt{1-x} \right)\frac{}{}\right]_0^1=\color{red}{\pi}. $$