Integration by parts and notation. I've just found that I must be missing something about the integral notation when it comes to the integration by parts.
First, $\int_a^b \! f(x) \, \mathrm{d}x$ is perfectly clear: $\mathrm{d}x$ is an infinitely small difference; it also syntactically designates the variable of the integration. 
Meanwhile, it's not a purely syntactic thing, since you can, e.g. substitute variables which involves meaningful operations with $\mathrm{d}x$ and you see $f(x)\mathrm{d}x$ as a differential.
Now, consider the integration-by-parts formula:
$$\int_a^bu(x)v'(x)\mathrm{d}x = uv |_a^b - \int_a^bv(x)u'(x)\mathrm{d}x$$
The thing is, that it's commonly written as:
$$\int_a^bu\mathrm{d}v = uv |_a^b - \int_a^bv\mathrm{d}u$$
Now, what I don't understand, is the strict meaning of $\mathrm{d}v$ on the left and $\mathrm{d}u$ on the right. Those don't designate the variable of the integration, since the ranges of the integration have not changed, and $u$ and $v$ are still functions of $x$. So, is that only some convenience trick and we're saying something like "yes, we know that it's not exactly correct, but as long as we know what we're doing it's ok", or there is some deeper non-syntactic meaning in replacing $\int_a^buv'\mathrm{d}x$ with $\int_a^bu\mathrm{d}v$?

If you check the calculus text you are reading it should explain that $u$, $du$, $v$ and $dv$ are defined to be $u=f(x)$, $du = f'(x)dx$, $v = g(x)$ and $dv = g'(x)dx$ (or any letters of your choice). Mathematicians are famously lazy and hide all sorts of pedantic notation in other simpler notation so they can spend their day doing math.
