Cheers for response - but it still seems to be turning out pretty hideous. Am I to reformulate 4p^2 in terms of u too (and of course get du in terms of dp) to get a full expression in terms of u to integrate with respect to u? It seems like that results in a simlarly nasty-looking product in need of integration.
I think now I'm beginning to discover exactly where one of the main missing chunks in my education in calculus lies. I think I understand what's going on here:
du/dp = 30p^2, therefore dp = du/30p^2
that cancels immediately with 4P^2 to produce 2/15, which can then be removed in front of the integral.
My weakness is that I never really got to grips with the significance of the d-terms, and what it was permissible to do with them. 'With respect to' is such a non-mathematical sounding idea, that I didn't really know what to do with 'with respect to u divided by 30p^2' - that seems like comparing apples and oranges to me, and hard to see as a seamless part of the mathematical expression. Oh well, thanks for the help anyway.