Split at $x=0$: $\int_{-1}^{0}(x+|x|)\,dx + \int_{0}^{2}(x+|x|)\,dx$. For $x\in[-1,0]$: $|x|=-x$, so $x+|x|=0$. Integral $=0$. For $x\in[0,2]$: $|x|=x$, so $x+|x|=2x$. $\int_0^2 2x\,dx = [x^2]_0^2 = 4$. Total $= 0+4 = \mathbf{4}$.