This is correct, though it took me a while to make sure of it :) In fact, if I had approached the problem in a better way initially then I would have gotten to this result without all the inverses.

Thanks everyone. I think I have solved it, and it doesn't require the inverse after all. @Robert 's answer led me to the solution by showing that I could construct the integral from a to b rather than some function of $u$ or its inverse. Once in that space, it is not too hard.

Yes, a very specific application -- I guess it may turn out that I have constructed the problem incorrectly, but I don't think so. I'll edit the question with more details as to the nature of the problem

Yeah, that's why I thought it might actually be impossible in a general sense. But you never know what some clever mathematician has come up with...

Ah, good point. I meant the former.