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how do I code

Integrate[u[x] D[v[x], x], {x, a, b}] 

to return the following straightforward undergraduate result?

u[b] v[b] - u[a] v[a] - Integrate[v[x] D[u[x], x], {x, a, b}]
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    $\begingroup$ You don't. In fact, you should perhaps abandon attempts to achieve that. Integration-by-computer is carried out very differently from work done by hand. $\endgroup$ – MarcoB Apr 4 at 17:48
  • $\begingroup$ Could always define your own little scheme, intbyparts[u_,dv_,a_,b_]=... if you feel up to the task. Need to make sure dv would be pretty easily integrable though. $\endgroup$ – Shinaolord Apr 4 at 20:23
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Use WolframAlpha

WolframAlpha["integration by parts (mathematical problem)", \
{{"FormulasPod:FamousMathProblem", 1}, "Content"}]

enter image description here

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