I found the short form of the formula for integration by parts at Wikipedia.

I think I'm finally starting to get it. The short form is:

Okay ... processing. I think this means that where there is a "u dv" you are actually multiplying variable u by variable dv. Also, dv is the derivative of v. (Why they didn't include a multiplier dot between u and dv is unknown. Edit: it is also unknown why the math script is taking the space out from between u and dv in the above.)

I'm still not sure if I have the correct intepretation here, because in every other case of d_ following an integration symbol the formula did not call for multiplication.

If I'm not mistaken, dx never means multiply, it's just this little thing that follows up integrations or antiderivations (presumably to make the multivariable calculus mavens happy). An example of dx not meaning "multiply by the derivative of x" is two lines up on the Wikipedia page (again "if I'm not mistaken").

I'm really jonesing to get this figured out. Got my fingers crossed.