Mathematics - Definite Integration Question with Solution | TestHub

$f(x)=x^2+{\int }_0^x{e}^{-t}f(x-t)dt=x^2+e^{-x}{\int }_0^x{e}^{t}f(t)dt$

$\Rightarrow f^{\prime }(x)=2x-e^{-x}\left(e^x\left(f(x)-x^2\right)\right)+e^{-x}\cdot e^{x}f(x)$

$\Rightarrow f^{\prime }(x)=2x+x^2\Rightarrow f(x)=\frac{x^3}{3}+x^2+K$

$$\begin{array}{c}\text{ But },f(0)=0\Rightarrow k=0\therefore f(1)=\frac{4}{3}\end{array}$$