(a)
Let’s show that for any integer \(n>1\) the Laplace Transform of \(f(t)=t^n\) satisfies \(\displaystyle \mathcal {L}(t^n)=\frac {n}{s}\mathcal {L}(t^{n-1})\) for \(s>0\), by applying the definition, \(\mathcal {L}(f)=\displaystyle \int _0^{\infty }f(t)e^{-st}\,dt\).
(b)
Let’s use the above, together with \(\mathcal {L}(t)=\frac {1}{s^2}\) to derive a formula or \(\mathcal {L}(t^n)\),
(c)
Let’s consider the piecewise continuous function \(f(t)=\left \{\begin{array}{lr} 4-t & \mbox { if }0\leq t<4\\ \\ 0 & \mbox { if }t\geq 4 \end{array}\right .\).
The graph of a piecewise continuous function. [Picture]
Figure 1: Graph of a piecewise continuous function

Let’s determine the Laplace Transform of \(f(t)\) by applying the definition,
\(\mathcal {L}(f)=\displaystyle \int _0^{\infty }f(t)e^{-st}\,dt\).