# 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.