Is the Alternative Method for Integration by Parts Simpler?

[DivergentSpectrum]

I have a question why everyone says
∫uv' dx=uv-∫u'v dx
why don't they replace v' with v and v with ∫vdx and say
∫uv dx=u∫vdx-∫(u'∫vdx) dx

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

 

[Mark44]

I have a question why everyone says
∫uv' dx=uv-∫u'v dx
why don't they replace v' with v and v with ∫vdx and say
∫uv dx=u∫vdx-∫(u'∫vdx) dx

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

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

 

[DivergentSpectrum]

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

right? its a lot 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 can't go wrong. so I am guessing for the sake of mathematical rigor they use the other form?

 

[Mark44]

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

right?

You checked it, didn't you?

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

I think of it like this: ∫v du = uv - ∫u dv. That's not very 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)

Ignore it in your intermediate work. Just add it at the end.

this would be wrong but as long as you keep in mind that the constant cooresponds to a vertical translation you can't go wrong. so I am guessing for the sake of mathematical rigor they use the other form?