5.4 Indefinite Integrals and the Net Change Theorem/43

$\int \limits _{-1}^{2}(x-2|x|)dx=\int \limits _{-1}^{0}(x-2(-x))dx+\int \limits _{0}^{2}(x-2(x))dx$

${\begin{aligned}&=\left({\frac {1}{2}}{x^{2}}+x^{2}\right){\bigg |}_{-1}^{0}+\left({\frac {1}{2}}{x^{2}}-x^{2}\right){\bigg |}_{0}^{2}\end{aligned}}$

${\begin{aligned}&=0-\left({\frac {1}{2}}(-1)^{2}+(-1)^{2}\right)+\left({\frac {1}{2}}(2)^{2}-(2)^{2}\right)-0&=\left({\frac {1}{2}}+1\right)+\left({\frac {1}{2}}(4)-4\right)\end{aligned}}$