Mathematics - Definite Integration Question with Solution | TestHub

Given,

$f\left(x\right)=x\left(1+{\int }_0^{1}f\left(t\right)dt\right)-{\int }_0^{1}tf\left(t\right)dt$

$f\left(x\right)=Ax-B ...\left(i\right)$

$A=1+{\int }_0^{1}f\left(t\right)dt=1+{\int }_0^1\left(At-B\right)dt$

$\Rightarrow A=2\left(1-B\right) ...\left(ii\right)$

Also $B={\int }_0^{1}tf\left(t\right)dt={\int }_0^1\left(A{\mathrm{t}}^2-Bt\right)dt$

$A=\frac{9}{2}B ...\left(iii\right)$

From $\left(ii\right), \left(iii\right)$

$A=\frac{18}{13}, B=\frac{4}{13}$

So, $f\left(x\right)=\frac{18}{13}x-\frac{4}{13}$

 $\Rightarrow f\left(6\right)=\frac{18\times 6-4}{13}=8$