0Before starting the discussion I would like to mention this problem
The conventional way to solve this problem is to deal with trigonometric identities and to mange by parts, but this type of problem can be easily solved by applying results of Beta and Gamma Functions.
———————————————————————————————————————————————————
Gamma Function:
The gamma function denoted by is defined for positive values of
by the integral
.......(1)
Now,
1. For any
2.
Integration by parts gives
As and
, the integreted part vanishes at both limits and therefore,
i.e.
3.
By direct computation converges
4.
Combining the above reletions,
Beta Function:
The beta function denoted by is defined for positive values of m and n by the integral
1.
This reletion can be established by giving the transformation
2.
Substituting
,then
Now letting , we have
also,
.
3.
This reletion can be established by by giving the transformation in the definition of Beta Function.
4.
This can be established by putting
in
5.
so,
—————————————————————————————————————————————————————
Now Lets solve the problem:
Now to solve this problem plug
in
=
As
