I have been learning the integration by parts formula.
$$\int u\ \mathrm dv = uv \ - \ \int v\ \mathrm du $$
I understand how the formula is derived when we keep everything in terms of $x$ (with $f(x)$ and $g'(x)$, etc.), but when we derive it using $u$ and $v$, we end up doing this at one point: $$\int u\frac{dv}{dx} \, \mathrm dx = \int u\ \mathrm dv $$ I don't quite understand how we go from the left hand to the right hand. Can someone help me with the intuition/reason we cancel out the $\mathrm dx$?
I have been taught to treat $\mathrm dy/\mathrm dx$, $\mathrm dv/\mathrm dx$, the $\mathrm dx$ in an integral, etc. as symbols and operators, not different parts of some fraction. Why are we cancelling the $\mathrm dx$ out, or why does it appear that way?