memo

para = U ->  UnitStep

qt = U[t] - U[t - a] /. para
Simplify[LaplaceTransform[qt, t, s]]

example 5.10 p.73

q[t_] := Sin[w t] U[t] /. para

qt = q[t] + q[(t - Pi/w)]

Simplify[LaplaceTransform[qt, t, s]]