In an article I've studying, I have $f$, a function with unit $[\text{length}]^{-1}$, and the exponential $e^{\int_0^x f(s)d(s)}$, where $x$ is a length.
This exponential is a real number.
I'd like to understand better how could I drop there the units, because they are important from physical meaning.
Many thanks!
When you integrate with $\mathrm{d}s$, you're integrating over a length. Because of that, the unit of the result "as an area [under the curve]" is [length]$^{-1}$ (from $f$) times [length] (from $\mathrm{d}s$, or from $0$ to $x$ as a definite integral), so the result is dimensionless as required.
Exponentials, logarithms, and other things that aren't really finite applications of addition, multiplication, or fixed powers cannot handle units, and must have dimensionless inputs. Trig functions accept "radians" as dimensionless units.