For any random variable (r.v.) $X$, you can write $X=X_+-X_-$, where $X_+:=\max(0,X)$ and $X_-:=\max(0,-X)$. Then you can write 
$$X_+=\int_0^\infty I\{X>x\}dx,\quad X_-=\int_0^\infty I\{-X>x\}dx,$$
and hence
$$X=\int_0^\infty (I\{X>x\}-I\{-X>x\})dx,$$
where $I\{\cdot\}$ is the indicator function.