The transform of $\bm{t^n}$

$$\begin{eqnarray*} \mathcal{L}[t^n](s) & = & \int_0^\infty {\rm e}^{-st} t^n {\r... ...ac{u^n}{s^{n+1}} {\rm d}u \\ & = & \frac{\Gamma(n+1)}{s^{n+1}} \end{eqnarray*}$$

where $\Gamma(\alpha) := \int_0^\infty {\rm e}^{-y} y^{\alpha - 1}\,{\rm d}y$ and $u := st$.