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Integration By Parts

  1. Sep 4, 2012 #1
    I understand this integration technique, for the most part. One thing I am curious to know is why, when you do your rudimentary substitution for this particular technique, does dv have to always include dx?
     
  2. jcsd
  3. Sep 4, 2012 #2

    HallsofIvy

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    "Integration by parts" is just the integral version of the "product rule" for derivatives:
    d(uv)= udv+ vdu. We can write that as udv= d(uv)- vdu and integrate both sides:
    [itex]\int udv= \int d(uv)- \int vdu[/itex]. Of course, [itex]\int d(uv)= uv[/itex].

    As for "does dv have to always include dx?", yes, of course. "dv" is a differential and you cannot have an "ordinary" function equal to a differential. A differential can only be equal to another differential.
     
  4. Sep 4, 2012 #3
    Is there a reason for this? Or have mathematicians defined this to be true?
     
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