Integration by parts form

1. Dec 4, 2014

DivergentSpectrum

I have a question why everyone says
∫uv' dx=uv-∫u'v dx
why dont they replace v' with v and v with ∫vdx and say
∫uv dx=u∫vdx-∫(u'∫vdx) dx

i think this form is alot simpler because you can just plug in and calculate, the other form forces you to think backwards and is unnecessarily complicated.

2. Dec 4, 2014

Staff: Mentor

Show us an example of how this would work, with say $\int xe^xdx$.

3. Dec 4, 2014

DivergentSpectrum

u=x
v=e^x
∫(x*e^x)dx=x*∫e^xdx-∫e^xdx
=xe^x-e^x+c

right? its alot simpler than thinking backwards and doing the substitution i always have to right it down my way cause the other way is too complicated.

edit: i just noticed that perhaps some people may be confused as to where the constant of integration goes
∫(x*e^x)dx=x*∫e^xdx-∫e^xdx
∫(x*e^x)dx=x*(e^x+c)-(e^x+c)
this would be wrong but as long as you keep in mind that the constant cooresponds to a vertical translation you cant go wrong. so im guessing for the sake of mathematical rigor they use the other form?

Last edited: Dec 4, 2014
4. Dec 4, 2014

Staff: Mentor

You checked it, didn't you?
I think of it like this: ∫v du = uv - ∫u dv. That's not very complicated.
Ignore it in your intermediate work. Just add it at the end.