Prove that $\displaystyle \int_{0}^{\pi} sin^n(x) ~dx = B( \frac{n+1}{2} , \frac{1}{2} )$

where n is a real number ( guys you can prove whether n can be complex ... )

Then use this result to solve a Physics problem :

Caculate the time taken for a particle to travel from the top of a unit semicircle to the bottom . ( under gravitational force )