Easy Laplace Transforms?

Question

.........{ 0, 0 < t < 1
f(t) = {
.........{ 2t - 2, t > 1

Plz ignore the "......."s! Plus : the three " { "s should be just one big " { ".

Question : Find (Laplace transform of) : L{f(t)}

Please help - I have no idea even where to start here!
Thanks.

Answer 1

Ordinarily the Laplace transform is the integral from 0 to infinity, but since f(t) = 0 for t < 1, we can take the integral from 1 to infinity. Then L(f) = integ{[e^(-st)]*(2t - 2)}dt. The antiderivative is (-2t/s + 4/s)*e^(-st). The limit as t -> inf is 0, and at t = 1, this is 4/s, so this is L(f), which is valid for s > 0.