Find initial value, given the Laplace transform

Question

Find the initial value ,

\begin{equation} F(s)=\frac{s+1}{(s+1)^2+10} \end{equation}

this is the question, I can implement the inverse laplace method but I do not know what to do at the last stage means after inversion. Now which method should I follow

Answer 1

No

Initial value theorem:

$$ \lim_{t \to 0} f(t) =\lim_{s \to \infty} s F(s)$$

Final value theorem:

$$ lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s)$$

Update

In your case, the initial condition, or $f(0)$ can be calculated from IVT (Initial Value Theorem):

$$ f(0)= \lim_{t \to 0^+} f(t) =\lim_{s \to \infty} s F(s)=\lim_{s \to \infty} \frac{s(s+1)}{(s+1)^2+10}=\lim_{s \to \infty} \frac{s^2}{s^2}=1$$

And the final result is:

$$f(0)=1$$

If you are curious about what is $f(t)$, here it is:

$$f(t)=\cos(\sqrt{10} t)\times e^{-t}$$