Integration By Parts

1. Sep 4, 2012

Bashyboy

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. Sep 4, 2012

HallsofIvy

"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:
$\int udv= \int d(uv)- \int vdu$. Of course, $\int d(uv)= uv$.

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. Sep 4, 2012

Bashyboy

Is there a reason for this? Or have mathematicians defined this to be true?