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?


Notice:

*

*When $x<0$:
$$\text{sign}(x)=-1$$

*When $x>0$:
$$\text{sign}(x)=1$$

*When $x=0$:
$$\lim_{x\to0^+}\text{sign}(x)=1$$
$$\lim_{x\to0^-}\text{sign}(x)=-1$$

\begin{align}
\int_{-1}^{2}\text{F}(x)\space\text{d}x &= \int_{-1}^{2}\text{sign}(x)\space\text{d}x \\ 
&= \int_{-1}^{0}\text{sign}(x)\space\text{d}x+\int_{0}^{2}\text{sign}(x)\space\text{d}x \\ 
&= \int_{-1}^{0}-1\space\text{d}x+\int_{0}^{2}1\space\text{d}x \\ 
&=-\int_{-1}^{0}1\space\text{d}x+\int_{0}^{2}1\space\text{d}x \\ 
&= -\left[x\right]_{-1}^{0}+\left[x\right]_{0}^{2}\\ &= -\left(0-\left(-1\right)\right)+\left(2-0\right)\\ &=-1+2 \\
&=1
\end{align}
