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...

This is correct right?

Assuming this is correct, from the product rule we have

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

So, where I'm a little stuck is why

I.e why does

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