MHB Definite Integral $\sqrt{\sin x}$: Answers

[juantheron]

$\displaystyle \int_{0}^{\frac{\pi}{2}}\sqrt{\sin x}dx.\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}dx =$

 

[JJacquelin]

Knowing the basic properties of the functions Beta and Gamma, it is easy to obtain :
first integral = (1/2)*Beta(3/4 ; 1/2)
second integral = (1/2)*Beta(1/4 ; 1/2)
and the product of the integrals = pi.