What is the best way to find the Inverse Laplace with Convolution Theorem?

Question

I have to calculate the inverse laplace using Convolution, but I get stuck when I have to integrate with the delta dirac. $$F(s)=\frac{s}{s^{2}+2s+2}$$

Answer 1

Hint: You don't need convolution, write $$F(s)=\frac{s}{s^{2}+2s+2}=\frac{s+1}{(s+1)^2+1}-\frac{1}{(s+1)^2+1}$$

Answer 2

Note: $$F(s)=\frac{s}{s^2+2s+1}=s\times\frac{1}{(s+1)^2+1}$$ but you cannot use convolution theorem on this as by definition: $$\mathscr{L}\left[f(t)\right]\ne s$$ and so it is difficult to prove this problem using convolution theorem like this, although this question is a good demonstration of shift theorem?

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$$F(s)=\frac{(s+1)}{(s+1)^2+1}-\frac{1}{(s+1)^2+1}$$ and since: $$\mathscr{L}\left[e^{at}f(t)\right]=F(s-a)$$ we can easily see that: $$\mathscr{L}^{-1}\left[\frac{(s+1)}{(s+1)^2+1}\right]=e^{-t}\mathscr{L}^{-1}\left[\frac{s}{s^2+1}\right]$$ and this makes finding $f(t)$ much easier