Solution
= Solution
Repeated integration by parts gives
$$
\mathcal L\{t^n\}(s)=\frac n s\mathcal L\{t^{n-1}\}(s),
\qquad \mathcal L\{1\}=\frac1s,
$$
so
$$
\boxed{\mathcal L\{t^n\}=\frac{n!}{s^{n+1}}}.
$$
Multiplication by $e^{at}$ shifts the transform:
$$
\boxed{\mathcal L\{e^{at}t^n\}(s)=\frac{n!}{(s-a)^{n+1}}}.
$$