Definite integral of sign function I need to calculte the integral of $F(x)=\text{sign}(x)$ (A partial function)
between $x=-1$ and $x=2$.
Of course we need to seperate the integral between $x>0$   and  $x<0$
but is it a case of improper integral ? or just seperate and calculate?

Once you've separated $\int_{-1}^2 F(x) dx$ it becomes by definition of the sign function (and because the value at exactly $x = 0$ doesn't matter)
$$
\int_{-1}^0 -1dx + \int_0^2 1dx
$$
which is easy to calculate and completely proper.
Integrals are very forgiving with discontinuities like this. You need quite a lot of them before they start causing theoretical problems with regards to properness and whether the integral is even defined. Technicality: As long as the set of discontinuities has measure $0$ (where "measure" is a more structured approach to the naïve notion of length / area / volume and so on), in particular if it's countable, then they do not hinder the Riemann integral from being defined.
