is there a cannonical meaning for $\int x\partial x$
the same way there is cannonical meanings for $\frac{\partial F}{\partial x}$, $\frac{dF}{dx}$ and $\int xdx$?
If so, how does this relate to the other two? What does it even mean?
EDIT: I am new to multivariable calculus and a bit confused about partial derivative vs normal derivative, though I know it has something to do with "local linearity" vs "local 'planarity'" (planarity comes with more dimensions, linearity allows for crossing out dx's, multiplication/addition nonsense of differential operators, ...)
If this is a case of forgetting a $dx$, such as when people write the incorrect expression $\int x=x^2/2+C$, we can take $\int dx\:x\partial_x$ to be the operator that sends $f$ to the antiderivatives of $xf^\prime$.