General Question: Say we have integral
$$ \int f(z)\ dz $$
Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its own right?
In case relevant:
In case relevant, what gave rise to this question was the claim that the integrand of $\int_\gamma p\ dx + q\ dy$ is an exact differential iff we can write the integrand as $dU = (\partial U / \partial x) \partial x + (\partial U / \partial y) dy$ for some function $U$. But that seems to suggest we can write
$$ U = p\ dx + q\ dy = (\partial U / \partial x) dx + (\partial U / \partial y) dy $$
And in this case I'm not sure what the object $p\ dx + q\ dy$ denotes (as opposed to the object $p+q$, which is quite straight forward).
"Integrand" only means "something that you have to integrate". It can be the function $f(z)$, or the differential form $f(z)\,dz$, depending on which point of view you prefer. An "exact differential" is a particularly nice differential form, so I guess that the point of view is the latter.
The object $p\,dx + q\,dy$ is exactly a differential form. Click on the links for more!