Im working out a set of Laplace inverses, from this one
$$\mathcal{L}^{-1}(\frac{-0.2}{s+5}+\frac{0.2s+0.8}{(s+2)^2+1})$$
I understand that the first term is
$-0.2e^{-5t}$ but the question is the second one, from what Ive been told this is $$0.2e^{-2t}cost+0.8e^{-2t}sint$$
but in a certain book Ive seen a concept called "generic decaying oscillatory" that shows the inverse Laplace transform of the form
$$\frac{Bs+C}{(s+a)^2+\omega^2}=e^{-at}(Bcos(\omega t)+\frac{C-aB}{\omega}sin(\omega t))$$
then it would be
$$0.2e^{-2t}cost+0.4e^{-2t}sint$$
How to justify the use of one or another? or what parameter use to choose?
Thanks
The relevant identity is $L^{-1}(F(s+a))=e^{-at} L^{-1}(F(s))$. So when you go to obtain the cosine term, you want it to look like $\frac{A(s+a)}{(s+a)^2+1}$. To get the "+a" you have to "add and subtract" a term, which changes the coefficient that would be contributing to the sine term. (This is similar to "completing thet square".) This ultimately results in the formula that you had in the middle.