Simple inverse laplace transform problem

Question

I have the follow excercise. I am aware of partial fraction expansion, but the roots are imaginary in this problem. Does somebody know how to solve it? Thanks.

$$ \mathcal{L} ^ {-1} (1 / (s ^ 2 + 4 s + 5)) $$

The roots of the denominator are $-2 + i$ and $-2 - i$, so I am stuck there.

Thanks.

Answer 1

Hint: $\dfrac{1}{s^2+4s+5}=\dfrac{1}{(s+2)^2+1}$. Now use the inverse Laplace transform of $\dfrac{1}{x^2+a^2}$, and notice that you need a "correction factor" of $e^{kt}$ for some $k$ "due to translation"