5.4 Indefinite Integrals and the Net Change Theorem/43

${\begin{aligned}\int _{-1}^{2}(x-2|x|)dx&=\int _{-1}^{0}(x-2(-x))dx+\int _{0}^{2}(x-2(x))dx=\int _{-1}^{0}3x\,dx-\int _{0}^{2}x\,dx\\[2ex]&=\left({\frac {3x^{2}}{2}}\right){\bigg |}_{-1}^{0}-\left({\frac {x^{2}}{2}}\right){\bigg |}_{0}^{2}\\[2ex]&=\left[{\frac {3(0)^{2}}{2}}-{\frac {3(-1)^{2}}{2}}\right]-\left[{\frac {(2)^{2}}{2}}-{\frac {(0)^{2}}{2}}\right]\\[2ex]&=\left[-{\frac {3}{2}}\right]-\left[{\frac {4}{2}}\right]\\[2ex]&=-{\frac {7}{2}}\\[2ex]\end{aligned}}$