I am trying to find the Laplace transform of $t^2$. My work is as follows:
$$\begin{align} \int_0^\infty e^{-st} t^2 \ dt &= \left[ t^2 \left( -\dfrac{e^{-st}}{s} \right) \right]_0^\infty - 2 \int_0^\infty t \left( -\dfrac{e^{-st}}{s} \ dt \right) \\ &= \dfrac{2}{s} \int_0^\infty e^{-st} \ dt \ \ \text{(Since the theory of Laplace transforms requires that $\Re(s) > 0$.)} \\ &= \dfrac{-2}{s^2} \int_{-s(0)}^{-s(\infty)} e^u \ du\\ &= \dfrac{-2}{s^2}e^{-s (\infty)} + \dfrac{2}{s^2} e^{-s(0)} \\ &= \dfrac{2}{s^2} \ \ \text{(Again, since $\Re(s) > 0$.)} \end{align}$$
Apparently, the solution is $\dfrac{2}{s^3}$. I would appreciate it if people could please point out where my error is.