The dxdx 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.
