I am note sure what you are doing.

Given the integral,

You can do it by parts,

and

Thus,

and

But the question is,

What is,

You can write,

Let,

then,

Thus,

You can let,

and

Thus,

and

Then, we need to find what,

is?

To do that we can write,

Let,

and

Then,

and .

Thus, by parts,

But then what is?

You can write,

Let, then .

Thus,

This is certainly doable!

The details, and the actual integral I leave to you do complete. (Maybe, there is an easier way, but this is one of the longest I seen).