How to find the inverse Laplace transform of $\frac{s}{(s+1)^2(s+2)}$?

Question

I would need a little help in finding the inverse Laplace transform of the function: $$f(s)=\frac{s}{(s+1)^2(s+2)}.$$

Thanks in advance.

Answer 1

Hint: recall that: $$ \mathcal{L}^{-1}\left(\frac{1}{s+a}\right)=e^{-ax},\qquad \mathcal{L}^{-1}\left(\frac{1}{(s+a)^2}\right) = xe^{-ax} $$ and apply a partial fraction decomposition.

Answer 2

Hint: the partial fraction decomposition could begin with something like:

$$ \frac{s}{(s+1)^2(s+2)} = \frac{2s+1}{(s+1)^2} + \frac{-2}{s+2} $$

Now you continue from here.