You're right. I think I know where is my error... Hmm no...
Let me show you what I did : Let , .
So the integral becomes .
Now let .
We know that and that . Furthermore, and .
Therefore we can rewrite the integral as , and from it I reached the result I gave in my first post. I thought I might have made an error for , but I don't think so. Can you help me finding where is my error?
Before I read I tried by integration by parts and then by substitution but didn't reach anything. Now following your trick, I got that the final result is...................................... or . But you know what? I've a strong feeling I made at least an error in all that.