= Solution
{c}
Write $I_n(p)=\int_0^\infty e^{-pt}t^n\,dt$. For $\operatorname{Re}p>0$, <exponential decay> dominates every fixed power of $t$, so the <improper integral> converges absolutely and $e^{-pT}T^n\to0$. For $n\geq1$, <integration by parts> gives
$$
I_n(p)=\left[-\frac{e^{-pt}t^n}p\right]_0^\infty
+\frac np\int_0^\infty e^{-pt}t^{n-1}\,dt
=\frac np I_{n-1}(p).
$$
The preceding part gives the base case $I_0(p)=1/p$. Repeated application, or <mathematical induction>, yields
$$
\boxed{\mathcal L\{t^n\}(p)=\frac{n!}{p^{n+1}},\qquad \operatorname{Re}p>0.}
$$
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