Determine the Laplace transform of the following function:
$$ f(t) = 3t^{4}e^{-2t} $$ When referencing our table of Laplace Transforms, we recognize that the function above resembles property #23 (shown below): $$ \mathcal{L}[t^ne^{at}] = \frac{n!}{(s-a)^{n+1}} $$ With this in mind, we proceed as follows: $$ \begin{align} \mathcal{L}[f(t)] & = F(s) \\ & = 3 \Big[ \frac{4!}{(s+2)^{4+1}} \Big] \\ & = 3\Big[ \frac{24}{(s+2)^5} \Big] \\ \end{align} $$
$$ F(s) = \frac{72}{(s+2)^5} $$
Continue on to Properties of the Laplace Transform example problem #2...