Integration By Parts

  • Thread starter Bashyboy
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Main Question or Discussion Point

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?
 

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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.
 
  • #3
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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.
Is there a reason for this? Or have mathematicians defined this to be true?
 

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