Question:

$$\int_{-1}^{2} x[x] dx = \int_{-1}^{0} x[x] dx + \int_{0}^{1} x[x] dx + \int_{1}^{2} x[x] dx$$ $$= \int_{-1}^{0} x(-1) dx + \int_{0}^{1} 0 dx + \int_{1}^{2} x(1) dx = -\int_{-1}^{0} x dx + \int_{1}^{2} x dx$$ $$= -\left(\frac{x^2}{2}\right)_{-1}^{0} + \left(\frac{x^2}{2}\right)_{1}^{2} = -\left(0 - \frac{1}{2}\right) + \left(\frac{4}{2} - \frac{1}{2}\right) = \frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2$$