I recently saw someone post an equation like this:
$$\int_a^b {dx \over f(x)}$$
Does this just mean $\int_a^b {1 \over f(x)} dx$? As far as I know, the $dx$ isn't an actual term and just notational convenience to declare the argument of the function that's integrated so we don't have to define it seperately.
Relatedly, assuming I have a function $f$, can I represent the antiderivative as $\int f$?
If you think of a definite integral for area as Leibniz did as an infinite sum of infinitely many infinitely thin rectangles each of which has base $dx$ then the position of $dx$ in the product "height times base" does not matter - but you do need it. The two expressions in the question are the same. See Why can't the second fundamental theorem of calculus be proved in just two lines? .
So $dx$ does more than simply indicate which variable you are integrating with respect to. That's really important in applications in physics.
When there is just one variable and you are more interested in theory than in applications you can in fact write $\int f$. Just remember that represents any of the infinitely many antiderivates (hence the "constant of integration").