Laplace Transform using t-shift

Question

$$f(t)=\begin{cases}cos(πt), & 1\leq t < 4 \\ 0, &elsewhere \end{cases}$$

Okay, I attempted to write it in terms of step functions and I got

$$ f(t) = cos(πt)u(t-1)-cos(πt)u(t-4)$$

But Now I'm not sure how to get the $$cos(πt)$$ for both parts in terms of (t-1) and (t-4) so then I could use the t-shift theorem. Can someone help me out?

Thanks!

Answer 1

Remember that $$\mathcal L(g(t)u(t-a)) = e^{-as}\mathcal L(g(t+a)),$$ where $\mathcal L$ is the Laplace transform. This is probably the version of the 't-shift' theorem you want to use here.