Laplace transform of piecewise function with 3 parts

Question

How to solve the question like this?

$f(x) = \begin{cases} 1 & 1\leq t <3 \\ te^{-3t} & 4\leq t < 5 \\ 0 & otherwise \end{cases} $

I need to consider 4 cases? Which is $0<t \leq 1$, $1 \leq t < 3$, $ 4 \leq t < 5 $ and $t > 5$?

Answer 1

Hint: Write your function in terms of the Heaviside Step Function, i.e, $f(t) = 1\,[u(t-1)-u(t-3)]+te^{-3t}\,[u(t-4)-u(t-5)]$.