This Formally gives
Now justifying the limits
Both of the above limits only exsist when
This turns the right hand side into a form of the Beta function!
Now using Euler reflection formula the product of the two Gamma functions can be written as
Are you expecting a beautiful solution ?
Intuitively , I think there must be another method which doesn't require any knowledges of Gamma function .
no, i'm not looking for a beautiful solution, i just wanted to share this problem.
here's my solution:
thus the integral equals