Annihilist wrote:I haven't learned how to do integration by parts. I'm leaving it to others in this forum, if anyone can be bothered.
Integration by parts is fun - it's just using the product rule for derivatives in reverse.
d(uv)/dx = vdu/dx + udv/dx
Therefore, vdu/dx = d(uv)/dx - udv/dx
Or in differential form,
vdu = d(uv) - udv
So the integral of vdu = uv - the integral of udv
To actually use this to do integrals can be a little tricky, since you have to find appropriate v and du/dx by guesswork. But with practice you soon get to recognise likely suspects.
Sometimes, the application of integration by parts doesn't lead to an integral that's immediately solvable but it does give you a relationship that can be used to solve (or at least simplify) the original integral.