I would like to show you a proof for the integral,
It is for the proving of my previous discussion,"On the Summation of Hypergeometric series".
Then by splitting the integral we get,
We have our integral but with an additional integral,for that lets use this integral.Since,
Then taking d.w.r.c,we get,
And hey! That's our required equation.So,
So the only thing left is to find the value of the derivative,which is quite easy once you have the knowledge of beta function and the diagamma function,
Then adding both derivative we have,
And as our corollary ,(I love corollary),
Where is Euler constant about 0.577.