Fourier transform of absolute values

Question

Is the following statement true? $$\int_{-2}^2 (2-|x|)e^{-iwx}\ dx=\int_{-2}^0 (x+2)e^{-iwx}\ dx + \int_0^2 (2-x)e^{-iwx}\ dx$$ This seems logical to me, but it does not give me the correct answer. Why is this the case and how do you properly evaluate such integral?