Use the definition of the Laplace transform to find L{f(t)} f(t) = {t, 0 ? t ? {2 Solution Definition - Laplace transform of a function f(t) is defined as follows L{f(t)} = integrate[0, inf] f(t)e-st dt So for 0 t 1, the questions says f(t) = t and you just plug it into the definition and get integrate[0, inf]{te-st}dt and evaluate the integral. First step of the intgral is integration by parts, let u=t and dv/dx = e-st , then integration by part formula says uv - integrate v(du/dx), which gives us [- (1/s) te-st] - integrate[0, inf] {-(1/s)e-st}dt. Note that the \"uv\" part [-(1/s) te-st] is equal to zero, and then you just have to integrate by parts the integral part again to get 1/s2 same thing for t 1 . plug in 2-t, and you get L{f(t)}=integrate[0, inf]{(2-t)e-stdt. Notice that the -t term you already did it in part a so you really just have to integrate 2e-st . which gives you 2/s.