Use of big mysterious shortcuts in academic papers, example integration by parts

Question

I read a paper where the author did a very strange but valid integration by parts:

What I thought was unusual is the repeating occurrence of $(q_T - q_t)$ in essentially all the terms (ignore $h$). Less unusually (sadly), I had absolutely no clue how the author obtained the expression.

After (embarrassingly) many hours, I figured out the author used a very unusual "definite integration by parts" shortcut:

$$ \int_a^b udv = -[(v(b)-v)u]\Big{|}_a^b + \int_a^b (v(b) - v) du $$

What I would like to do is not be so troubled by such shortcuts in the future, and would greatly appreciate your advice and knowledge regarding the use of such shortcuts in academic papers, etc.

• Is this a well-known expression somewhere or in some field?

• Or is this just a bit of hidden manipulation to get nice aesthetic properties?

• Is there an expectation that the reader won't be confused by the use of such a shortcut?

• Is it a bad sign that it confused me (a grad student) so much and took me a few hours to get around it?

Thanks for your kind responses.

Answer 1

Sadly, there is no one recipe for addressing this issue. In my own personal experience, the depth of explanation is generally not the fault of the author but the restrictions of the Journal in how many pages/words are permitted. As such, you will very often see compressed working to accomodate such restrictions.

The best approach is to email the authors directly. I know that when I've received questions I'm more than happy to respond. And when I've asked questions, I've had nothing but positive experiences in the responses I receive.

Answer 2

It's not a secret trick at all. If I ask you to choose $u,\,v$ so that an integrand is $uv^\prime$, after fixing $u$ your choice of $v$ isn't unique because a constant can be added to it. Let $v_0$ denote the choice for $v$ that was in your head; the author was thinking of $v_0-v_0(b)$ instead. In other words, they made $v$ unique with the convention $v(b)=0$.

Usually $v$ is chosen either to vanish at one end or the other, or to be "the obvious" option (e.g. $x^2$ instead of $x^2+5$, regardless of the integration limits.) When an author doesn't tell you what they did, check those three options until one makes sense. Usually the upper limit won't be used to "calibrate" $v$ in the way it was here, although I think financial analysis might be exceptional in that regard because of how often the at-$T$ behaviour matters.