Integration by Parts: Understanding dv & dx

[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?

 

[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.

 

[Bashyboy]

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?