Integration By Parts - Proof

Hi Guys,

I've had a search for "Integration by parts" and couldn't quite find what I was looking for... so apologies if this is actually somewhere in the forum and I've just missed it.

My Question has two parts.

Firstly a (possibly) silly sanity check...

$\displaystyle \int dx/dy = y + C$

This is correct right?

Assuming this is correct, from the product rule we have

$\displaystyle dy/dy = v.du/dx + u.dv/dx$

So if we integrate both sides we have

LHS = y = uv

But I'm puzzled by the integration of the RHS

The proof I'm reading gives

$\displaystyle RHS = \int u.dv/dx.dx + \int v.du/dx.dx

= \int u.dv + \int v.du$

So, where I'm a little stuck is why

$\displaystyle \int u.dv/dx.dx \neq uv$

I.e why does

$\displaystyle \int dv/dx.dx \neq v?$

Any help would be much appreciated. Sorry for the rubbish latex btw, most of the little image things under the latex cheat sheet haven't loaded.

Thanks in advance!

James