$\begin{align}  & I=\int\limits_{-1}^{1}{({{x}^{2}}-3x+2)dx-}\int\limits_{1}^{2}{({{x}^{2}}-3x+2)dx+}\int\limits_{2}^{4}{({{x}^{2}}-3x+2)dx} \\  & =\left. \left( \frac{{{x}^{3}}}{3}-\frac{3{{x}^{2}}}{2}+2x \right) \right|_{-1}^{1}-\left. \left( \frac{{{x}^{3}}}{3}-\frac{3{{x}^{2}}}{2}+2x \right) \right|_{1}^{2}+\left. \left( \frac{{{x}^{3}}}{3}-\frac{3{{x}^{2}}}{2}+2x \right) \right|_{2}^{4}=\frac{19}{2} \\ \end{align}$