Integration: u substitution problem I'm trying to integrate the following:
$\int_{0}^{\infty} x^2 e^{-2x}$
I know that $\int_{0}^{\infty} x^n e^{-x} dx = n!$
Feels like a $u$ substitution problem but I'm having trouble making use of the above.
Thanks,
Mariogs

Look at the more general integral, related to Laplace transforms, of this type:
\begin{align*}
\mathcal{L}\left\{t^n\right\}\left(s\right)&=\int_0^\infty t^n e^{-st}\:dt,
\end{align*}
and you may solve using integration by parts to always get something of the form
\begin{align*}
\int_0^\infty t^n e^{-st}\:dt & = \frac{n!}{s^{n+1}}.
\end{align*}
In terms of the Gamma function you wrote,
\begin{align}
\int_{0}^\infty t^ne^{-st}\:dt&=\frac{\Gamma\left(n+1\right)}{s^{n+1}},
\end{align}
which looks somewhat nicer.
