I already know the answer but I'm searching for an understanding or explanation of why it's the answer.
The question is: find the inverse Laplace transform of
$$X = \frac{s+1}{(s+1)^2+1} $$
answer is $$x = \exp(-t)\cos(t)$$
my professor pretty much just said look at the laplace table and mix the $\frac 1 {s+a} = \exp(-at)$ and $\frac{s}{s^2+w^2} = \cos(wt)$ rows together because they 'look' similar to the equation in the problem. This explanation doesn't give me much understanding of why you get that answer or a concrete method to follow so i can answer similar questions. Hope you can give me some insight!
Note that $$ \frac{s+1}{(s+1)^2+w^2}$$ is $F(s+1)$ where $F(s) = \frac{s}{s^2+w^2}$ is the inverse laplace transform of $\cos (\omega t)$.
Due to the shift of $s\to s+1$ we get a factor of $e^{-t}$ in front of $ \cos (\omega t)$
Thus the final answer is $$ e^{-t}\cos (\omega t)$$