$$\int_1^3f(x)\,\mathrm dx=\int_1^{\operatorname e}\frac{\mathrm dx}x+\int_{\operatorname e}^3x\,\mathrm dx=\log x\bigg\vert_1^{\operatorname e}+\tfrac12x^2\bigg\vert_{\operatorname e}^3$$$$\quad=\left(\log{\operatorname e}-\log1\right)+\tfrac12\left(3^2-{\operatorname e}^2\right)=\tfrac12\left(11-{\operatorname e}^2\right).$$Gruß
