Prove that
This Problem is not as difficult as it looks. Use the property :
The first integral equals -1.
Let -------(1)
A property of definite integral says that
therefore
--------(2)
Add (1) and (2):
(from Euler's Reflection Formula! )
Recall that .
Therefore
Here is another way to evaluate the integral. Write it out as a Riemann sum, the old-school way:
So
This exactly fits the format for the multiplication theorem for the gamma function! So