Trouble Understanding Replacement for Radii in Proof Problem

[batman394]

THis is more of a proof problem. I worked the problem up until the point where it begins a replacement for r/R.

The book has enough examples to understand it up until that point.. but then it jumps from radii to simply the n term.

I got a copy of the solution from the professor and I just don't get why they do that. I can't integrate that function in any way to get that new n-based equation.

Any ideas on what's going on? I can't find anything in the book on that.

 

[batman394]

so we've sort of got it figured out.. the inner radius is r. the outer is R. the n in the system is just the ratio of the centerline velocity, and average velocity. relate that to the r and R.
and you can have n = r/R = u(bar) / u(c) ... and do a replacement and then do integration by parts.. and it all works out.. eventually.. the intergration by parts was a pain.