Mathematics - Definite Integration Question with Solution | TestHub

$F(t)={\int }_0^{t}f(t-y)g(y)dy$ $={\int }_0^t{e}^{t-y}ydy=e^t{\int }_0^t{e}^{-y}ydy$ $=e^t{\left[-y{e}^{-y}-e^{-y}\right]}_0^t=-e^t{\left[y{e}^{-y}+e^{-y}\right]}_0^t$ $=-e^t\left[t{e}^{-t}+e^{-t}-0-1\right]=e^t\left[\frac{t+1-e^t}{e^t}\right]=e^t-(1+t)$