Solution
= Solution
Part (i) gives $\mathcal L(t)=1/s^2$ and $\mathcal L(te^{-t})=1/(s+1)^2$. The <convolution theorem for Laplace transforms> therefore gives
$$
f(t)=\int_0^t(t-v)ve^{-v}\,dv=t\int_0^tve^{-v}\,dv-\int_0^tv^2e^{-v}\,dv.
$$
Repeated <integration by parts> yields
$$
\int_0^tve^{-v}\,dv=1-(t+1)e^{-t},\qquad\int_0^tv^2e^{-v}\,dv=2-(t^2+2t+2)e^{-t}.
$$
Substitution gives \b[$f(t)=t-2+(t+2)e^{-t}$], agreeing with the <contour integration> result.