\(\blacksquare \) Remember that \(u_{c}\left ( t\right ) f\left ( t-c\right ) \Longleftrightarrow e^{-cs}F\left ( s\right ) \) and \(u_{c}\left ( t\right ) f\left ( t\right ) \Longleftrightarrow e^{-cs}\mathcal {L}\left \{ f\left ( t+c\right ) \right \} \). For example, if we are given \(u_{2}\left ( t\right ) t\), then \(\mathcal {L}\left ( u_{2}\left ( t\right ) t\right ) =e^{-2s}\mathcal {L}\left \{ t+2\right \} =e^{-2s}\left ( \frac {1}{s^{2}}+\frac {2}{s}\right ) =e^{-2s}\left ( \frac {1+2s}{s^{2}}\right ) \). Do not do \(u_{c}\left ( t\right ) f\left ( t\right ) \Longleftrightarrow e^{-cs}\mathcal {L}\left \{ f\left ( t\right ) \right \} \) ! That will be a big error. We use this allot when asked to write a piecewise function using Heaviside functions.