<s>[INST] Calculating an inverse Laplace transform I have a pain for a problem... need to find the inverse Laplace transform of:
$$\frac{s^2+s+1}{(s+1)(s+2)^2(s^2+4s+9)}$$
Now, I get that we have to expand the partial fraction, I managed to get to this:
$$\frac{A}{(s+1)}+\frac{B}{(s+2)}+\frac{C}{(s+2)^2}+\frac{Ds+E}{(s^2+4s+9)}$$
and after doing some expanding and simultaneous equations, I got:
$A=\frac{5}{6}$; $B=\frac{64}{5}$; $C=\frac{-67}{5}$; $D=\frac{-289}{30}$; $C=\frac{-347}{10}$
So now, I have
$$\mathcal{L}[f(t)]=\frac{5}{6}\frac{1}{(s+1)}+\frac{64}{5}\frac{1}{(s+2)}-\frac{67}{5}\frac{1}{(s+2)^2}-\frac{289}{30}\frac{s}{(s^2+4s+9)}-\frac{347}{10}\frac{1}{(s^2+4s+9)}$$
from here I use the table of inverses in the textbook to get:
$F(s) = \frac{5}{6}e^{-t}+\frac{64}{5}e^{-2t}-\frac{67}{5}te^{-2t}+\dots$
and then I'm stuck on that last polynomial... I have looked and looked but found nothing... and I'm not even sure I did the first part correctly.
Please help!
 [/INST] Hint. Note that
$$s^2+4s+9=(s+2)^2+(\sqrt{5})^2$$
and take a look at "exponentially decaying
sine/cosine wave" in this Table.
P.S. Check your partial fraction decomposition. It should be
$$\frac{1/6}{(s+1)}-\frac{3/5}{(s+2)^2}-\frac{s/6-1/10}{(s^2+4s+9)}
=\frac{1/6}{(s+1)}-\frac{3/5}{(s+2)^2}-\frac{(s+2)/6-13/30}{(s+2)^2+(\sqrt{5})^2}.$$
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