Is there any way to simplify $\int udv-\int vdu$?
Something related is that $\int udv+\int vdu$ can clearly be simplified to $uv$, and I am wondering if it can be done for $\int udv-\int vdu$
Simplify $\int udv-\int vdu$
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real-analysis
integration
analysis
integration-by-parts
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0Consider $\frac uv$. It's not correct, but it's close. Maybe it leads to an answer. – 2017-01-22
2 Answers
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We know that by the integration by parts formula we have $$\int u dv = uv -\int v du $$ $$\Rightarrow \int u dv -\int v du =uv -2\int v du = 2\int u dv -uv $$ I don't think there exists a nice simplification of the result. Hope it helps.
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0This answer *proves* that there cannot exist any "nice simplification of the result". – 2017-02-25
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Every expression like $df=f'(x)dx$ called Exact Differential (Wolfram) that your example is an exact differential $$d(uv)=u\,dv+v\,du$$ Some exact differentials you may use to simplify expression like $u\,dv-v\,du$ are $$d\Big(\frac{v}{u}\Big)=\frac{u\,dv-v\,du}{u^2}$$ $$d\Big(\arctan\frac{v}{u}\Big)=\frac{u\,dv-v\,du}{u^2+v^2}$$ $$d\Big(\ln\frac{v}{u}\Big)=\frac{u\,dv-v\,du}{u\,v}$$ In these cases you need to multiply an extra expression which convert yours to one exact differentials.