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