Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{rl}& {\int }_0^2{\mathrm{xF}}^{\prime }(\mathrm{x})\mathrm{dx}=6\\ & ={\mathrm{xF}(\mathrm{x})|}_0^2-{\int }_0^2\mathrm{f}(\mathrm{x})\mathrm{dx}=6\\ & =2\mathrm{F}(2)-{\int }_0^2\mathrm{xF}(\mathrm{x})\mathrm{dx}=6[\therefore \mathrm{f}(2)=2\mathrm{F}(2)=2]\\ & {\int }_0^2\mathrm{xF}(\mathrm{x})\mathrm{dx}=-\mathrm{2...}(1)\\ & \Rightarrow {\int }_0^2\mathrm{F}(\mathrm{x})\mathrm{dx}=-\mathrm{2...}(2)\end{array}$

Also
${\int }_0^2{x}^2{F}^{\prime \prime }(x)dx={x^2{F}^{\prime }(x)|}_0^2-2{\int }_0^2{x}^{\prime }{F}^{\prime }(x)dx=40$
$\begin{array}{rl}& =4F^{\prime }(2)-2\times 6=40\\ & F^{\prime }(2)=13\\ & \therefore F^{\prime }(2)+{\int }_0^{2}F(x)=13-2=11\end{array}$