I know that I cannot use integration by parts since it always loops.
Is this your integral: ??
If so you can use integration by parts. The key to this one is that you apply integration by parts twice. After you apply it twice, you'll see the original integral reappear.
I.e. you'll see after applying IBP twice that you will end up with
Then you treat as a variable and then solve for it.
Does this make sense?
Hello, lord12!
Yes, it "loops" when doing it by-parts, but that's what we want!
Let. Why did they use such an ugly fraction?
We have: .
. . By parts: .
And we have: .
. . By parts: .
And we have: .
. . . . . . . . . .
We have: .
Factor: .
Then: .
Therefore: .