Usage of dx in Integrals All the integrals I'm familiar with have the form:

$\int f(x)\mathrm{d}x$.

And I understand these as the sum of infinite tiny rectangles with an area of: $f(x_i)\cdot\mathrm{d}x$.
Is it valid to have integrals that do not have a differential, such as $\mathrm{d}x$, or that have the differential elsewhere than as a factor ? Let me give couple of examples on what I'm thinking of:

$\int 1$

If this is valid notation, I'd expect it to sum infinite ones together, thus to go inifinity.

$\int e^{\mathrm{d}x}$

Again, I'd expect this to go to infinity as $e^0 = 1$, assuming the notation is valid.

$\int (e^{\mathrm{d}x} - 1)$

This I could potentially imagine to have a finite value.
Are any such integrals valid? If so, are there any interesting / enlightening examples of such integrals?

No, it's not valid. The dx in the integral is a representation of the fact that the integral is obtained as an area, so multiplying the "average" of the function value at each point by an infinitesimal interval.
As the manner in which we don't calculate the area does not change, the notation does not change.
There are different notations that are used when the integral is over a curve, or over more than variable (thus leading for example to volumes).
The d(variable) notation is also used as a reminder that the integral is against a specific variable and not another, e.g. that int x/y dx differs from int x/y dy.
