The given problem asks to find the Inverse Laplace Transform of:

$\frac{s}{(s+2)^2 + 4}$

This expression resembles the standard form for the Laplace transform of a cosine function with a phase shift. The general formula for the inverse Laplace transform of:

$\frac{s - a}{(s - a)^2 + b^2}$

is:

$e^{at} \cos(bt)$

Steps to Solve:

1. First, notice that the given expression can be written in a form similar to the above template:

   $\frac{s}{(s + 2)^2 + 2^2}$

   Here, $a = -2$ and $b = 2$.

2. Using the formula, the inverse Laplace transform is:

$\mathcal{L}^{-1} \left( \frac{s}{(s + 2)^2 + 4} \right) = e^{-2t} \cos(2t)$

Final Answer:

The inverse Laplace transform of $\frac{s}{(s + 2)^2 + 4}$ is:

$e^{-2t} \cos(2t)$

Would you like more details on any step, or do you have any questions?

Here are 5 related questions for further exploration:

1. What is the inverse Laplace transform of $\frac{1}{s^2 + 4}$?
2. How do you handle inverse Laplace transforms involving exponential terms?
3. What are the properties of the Laplace transform that make solving such problems easier?
4. How do you apply partial fractions to more complex Laplace transform problems?
5. What is the relationship between Laplace transforms and differential equations?

Tip: Always check if your Laplace transform expression matches a standard form before solving, as this simplifies the process.