Where does integration by parts come from?

The best way to do well in maths, is to learn where things come from. Mastering the basics makes everything else much easier!
To start, consider the product rule with two functions u(x) and v(x) (using dashes to represent derivatives. e.g. y' is equal to dy/dx):
-> (uv)' = uv' + u'v
Now integrate both sides with respect to x:
-> ∫(uv)' dx = ∫uv' + u'v dx -> ∫(uv)' dx = ∫uv' dx + ∫u'v dx
Now look closely at the left hand side of the equation. The integral and the derivative cancel each other out, so the term can now simply become uv :
-> uv = ∫uv' dx + ∫u'v dx
We can now rearrange the equation into something that will be useful to us:
-> ∫uv' dx = uv - ∫u'v dx
This is the integration by parts formula, we can now go over how to use this derivation to solve integration problems such as: ∫ 3x e^3x dx
(Tip: be careful what you choose as u and v, otherwise you could make life much harder for yourself!)