Determine tha Laplace transform using Heaviside fucntion

Question

I want to determine the Laplace transform of the following function: $$f(t):t \mapsto \begin{cases}0, \quad t< 0 \\t, \quad 0\leq t \leq2\\2,\quad t>2\end{cases}$$

I have done it using standard integral transformation, however I wonder what is the method of determining Laplace transforms of such functions using Heaviside: $$H(t):t \mapsto \begin{cases}0, \quad t< 0 \\1, \quad t\ge 0\end{cases}$$

Answer 1

Note that, $$ f(t) = t[H(t) - H(t - 2)] + 2H(t - 2) $$ and

Answer 2

Here is one way: Let $f(t) = \int_0^t (H(\tau) - H(\tau-2)) d \tau$.

Then ${\cal L} f(s) = {1 \over s} ({\cal L} H(s)) (1-e^{-2s}) = {1 \over s^2} (1-e^{-2s})$.