Is it possible to integrate (t^2e^-t)/(t + 1)??
$\displaystyle \int t^2 e^{-t} (t+1)^{-1} $
$\displaystyle \int t^2 e^{-(t+1) + 1} (t+1)^{-1} $
$\displaystyle e \int t^2 e^{-(t+1) } (t+1)^{-1} $
Let $\displaystyle u = t + 1 $
$\displaystyle e \int (u-1)^2 e^{-u } (u)^{-1} $
$\displaystyle e \int \frac{ u^2 - 2u + 1 }{ ue^u }$
$\displaystyle e \int \frac{ u}{ e^u } - 2e \int e^{-u} + e \int (ue^u)^{-1} $
All of the above can be evaluated fairly easily. Use By Parts for most of them.
I lied, we cannot use by parts on that particular integral. It cannot be evaluated in elementary functions: Wolfram Mathematica Online Integrator
It's defined as $\displaystyle Ei(-x) $
So it would appear as if I have taken you down a path where you can integrate the integrand but not in terms of elementary functions. I'm sure an alternative approach will yield a result that can be fully explained in terms of elementary functions (i'll think of another way to start it off).
Calling...Simplependulum!!