The Beta Gamma Function

Before 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.

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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,

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 Now Lets solve the problem:

Now to solve this problem plug     

in

                                                  =

As