# Laplace transform of step function

1. Jun 19, 2012

### ElijahRockers

1. The problem statement, all variables and given/known data

Find laplace trasform of g(t)
g(t) = 0 for 0<t<1, t^2 for t>1

So this can be re-written as g(t) = t2*u1(t), where u is the unit step function.

By using the fact that L(uc(t)f(t-c)) = e-csF(s) i am trying to take the laplace...

So in this case, f(t-c) = t2 .... so then does f(t) = (t+1)2?

if so, I get L(f(t)) = e-s*L(t2+2t+1) which I can evaluate to get

e-s(2/s3 +2/s2 +1/s)

but i'm not sure if that whole shifting function thing I did is correct, specifically, f(t)=(t+1)2. the book's example uses a trig function that is already shifted so im a little unsure of how to proceed

Thanks.

Last edited: Jun 19, 2012
2. Jun 19, 2012

### vela

Staff Emeritus
Probably the easiest way to rewrite the function is to let u=t-1, so that t=u+1 and
$$t^2 = (u+1)^2 = (t-1)^2 + 2(t-1) + 1$$ which is what you had. You could also check your answer by evaluating the integral
$$\int_1^\infty t^2 e^{-st}\,dt$$ and see if you get the same answer.