Mathematics - Definite Integration Question with Solution | TestHub

$\int {\mathrm{xf}}^{\prime \prime }(2\mathrm{x})\mathrm{dx}=\mathrm{x}\frac{{\mathrm{f}}^{\prime }(2\mathrm{x})}{2}-\int \frac{{\mathrm{f}}^{\prime }(2\mathrm{x})}{2}\mathrm{dx}=\mathrm{x}\frac{{\mathrm{f}}^{\prime }(2\mathrm{x})}{2}-\frac{\mathrm{f}(2\mathrm{x})}{4}$

${\int }_0^1{\mathrm{xf}}^{\prime \prime }(2\mathrm{x})\mathrm{dx}={\left[\frac{{\mathrm{xf}}^{\prime }(2\mathrm{x})}{2}-\frac{\mathrm{f}(2\mathrm{x})}{4}\right]}_0^1=\left(\frac{{\mathrm{f}}^{\prime }(2)}{2}-\frac{\mathrm{f}(2)}{4}\right)-\left(0-\frac{\mathrm{f}(0)}{4}\right)$

$$\left(\frac{5}{2}-\frac{3}{4}\right)-\left(0-\frac{1}{4}\right)=2$$