A definite integral problem related to sliding down frictionless QUARTER CIRCLE
Solution 1:
You will not have any solution in terms of elementary functions. Instead you will obtain an elliptic integral.
Render $y =R\cos\theta$, then the integral becomes
$R^{-1/2}\int_0^{\pi/2}\dfrac{d\theta}{\sqrt{\cos\theta}}$
Next put in $\cos\theta=1-2\sin^2(\theta/2)$ and compare with the complete elliptic integral of the first kind described in Wikipedia.