"You are almost right except u and v are functions of x...if they were different variables you would have to assume one is a constant...making the integration exceedingly simple"

Okay, that may be where I'm going wrong. When I have taken u and v in the past, one of them is almost always a constant, yielding a derivative of zero. While the integration didn't seem exceedingly simple, the answers have usually been wrong.

Back to the drawing board.