Hello,

I guess there's a little typo in red, though not important for your further working

Well, actually, there's a problem, as for the convergence of the integral. But I'm a bit tired to study it ><=

let u=

Then du= dy

=

=

Is this right so far? Is there some reason why should be one? Could somebody show me how to integrate this by parts? (I know I should be able to do it but I am really struggling).

Make the substitution , assuming that . Otherwise, you'd have to change the boundaries of the integral, and it makes it more complicated.

Then, you'll get something like

For this integral, see the Gamma function : Gamma function - Wikipedia, the free encyclopedia and it should be okay !

I hope that what I babbled will help you in your process. Anyway, it's nice to see your working