Finding the Laplace transform of:
$$f(t) = \begin{cases}1 ,& 0 \leq t < 1 \\t^2 ,& 1 \leq t \leq 2 \\ 4, & t \ge 2\end{cases}$$
Heaviside /Unit step function: $$1+ (t^2 -1)u(t-1) + (4-t^2)u(t-2)$$
The Laplace transform of it is:
$$\mathcal{L}{(1)} = 1/s$$ $$\mathcal{L}{(t^2 -1)u(t-1)} = e^{-s} \mathcal{L}{f(t)} = e^{-s} \mathcal{L}{(t^2)} = e^{-s}\frac {2}{s^3}$$ $$\mathcal{L}{(4 - t^2)u(t-2)} = e^{-2s} \mathcal{L}{f(t)} = e^{-2s} \mathcal{L}{(??)} =...$$
Could someone verify if first of all my equation is right and how do I take the Laplace transform of the last equation.