The #dx# is there, for one, by convention. Recall that the definition of definite integrals comes from a summation that contains a #Deltax#; when #Deltax->0#, we call it #dx#. By changing symbols as such, mathematicians imply a whole new concept - and integration is indeed very different from summation.
But I think the real reason why we use #dx# is to clarify that you are indeed integrating with respect to #x#. For example, if we had to integrate #x^a#, #a!=-1#, we would write #intx^adx#, to make it clear that we are integrating with respect to #x# and not to #a#. I also see some sort of historical precedent, and perhaps someone more versed in mathematical history could expound further.
Another possible reason simply follows from Leibniz notation. We write #dy/dx#, so if #dy/dx=e^x#, for example, then #dy=e^xdx# and #y=inte^xdx#. The #dy# and #dx# help us keep track of our steps.
However, at the same time I do see your point. To someone with more experience than average in calculus, #int3x^2# would make as much sense as #int3x^2dx#; the #dx# in those situations is a bit redundant. But you can't expect only those people to look at the problem; students starting out in the subject are more comfortable with a little more organization in the problem (at least from my experience), and I think the #dx# provides that.
I am positive there are other reasons why we might use #dx# so I invite others to contribute their ideas.
