Find $\int_0^1 \sqrt{\frac{1-t}{t(1-0.5t)}}dt$. Find $\int_0^1 \sqrt{\frac{1-t}{t(1-0.5t)}}dt$.
I think that we can not use integration by parts and partial fractions. Is it possible to use change of variable and properties of definite integral? Or any other method? Thanks.

$$I=\int_{0}^{1}\sqrt{\frac{1-t}{t(1-t/2)}}\,dt = 2\sqrt{2}\int_{0}^{1}\sqrt{\frac{1-u^2}{2-u^2}}\,du=2\sqrt{2}\int_{0}^{1/\sqrt{2}}\sqrt{\frac{1-2v^2}{1-v^2}}\,dv $$
$$ I = 2\sqrt{2}\int_{0}^{\pi/4}\sqrt{\cos(2\theta)}\,d\theta = \sqrt{2}\int_{0}^{\pi/2}\left(\cos\varphi\right)^{1/2}\,d\varphi = \frac{4\pi\sqrt{\pi}}{\Gamma\left(\frac{1}{4}\right)^2}$$
by Euler's Beta function and the reflection formula for the $\Gamma$ function.
