Can you explain where the "Integration by parts" formula comes from?

Sure. If you remember how to calculate d/dx(uv) then you can understand how integration by parts works. d/dx(uv) = u(dv/dx) + v(du/dx). we can re-arrange this: u(dv/dx) = d/dx(uv) - v(du/dx). Now integrating both sides: |u.dv = uv - |v.du (Where I've used "|" for the integration sign) which is the integration by parts formula.

All you need to do is work out what you use as "u" and "dv", which comes down to experience.