# Laplace transform of t^2e^(at)??

Laplace transform of ${t}^{2}{e}^{at}$??
try to prove that
$\mathcal{L}\left\{{t}^{2}{e}^{at}\right\}=\frac{2}{\left(s-a{\right)}^{3}}.$
I've gotten to the last integration by parts where
$\underset{n\to \mathrm{\infty }}{lim}{\int }_{0}^{n}\frac{1}{\left(a-s{\right)}^{2}2{e}^{\left(a-s\right)t}}dt={\underset{n\to \mathrm{\infty }}{lim}\frac{2}{\left(a-s{\right)}^{3}}{e}^{\left(a-s\right)t}|}_{0}^{n}.$
Now what do I do?
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Prezrenjes0n
Evaluating the integral for a fixed n gives
$\frac{2}{\left(a-s{\right)}^{3}}{e}^{\left(a-s\right)n}-\frac{2}{\left(a-s{\right)}^{3}}{e}^{0}$
Assume that s>a and let n go to infinity. Then since $\left(a-s\right)n\to -\mathrm{\infty }$, the first term disappears and the limit is
$-\frac{2}{\left(a-s{\right)}^{3}}{e}^{0}=-\frac{2}{\left(-1{\right)}^{3}\left(s-a{\right)}^{3}}=\frac{2}{\left(s-a{\right)}^{3}}$