Simple question about direct integration

[Tuneman]

if I have a equation like this

I know I can integrate so that

but this does not make sense to me because I could think of many U's that if you were to take the derivitive with respect to the squiggly varibale, still includes both variables

 

[Hurkyl]

If the derivative with respect to xi includes both variables, then how does it look when you differentiate with respect to both xi and to eta?

 

[Tuneman]

http://www.threepointstance.net/tpsbb/images/smilies/clever.gif