if L{e^(at)} = 1/(s-a) and L{t} = 1/ s^2 then inverse L{ 1/(s-2)^2 } = f(t) = te^(at) There's a rule for it (a better way to put it). I have to go, though. If nobody posts it, put it up later.
Oops, I see your problem. You already thought of the partial fraction expansion, but that doesn't work for the degeneracy. The above from silvrstring is correct: you can put the following in your table. It's needed for degeneracy cases with partial fraction expansions. transforms to with region of convergence Re{s} > transforms to with region of convergence Re{s} <