To find the inverse Laplace transform of $\frac{1}{(s+2)^3}$, we can follow these steps:

Step 1: Recognize the standard form

We recognize that $\frac{1}{(s + a)^n}$ is a standard Laplace transform of a time-domain function. Specifically, the inverse Laplace transform of $\frac{1}{(s + a)^n}$ is related to $t^{n-1} e^{-at}$ (which is derived from the general formula for the inverse Laplace of powers of $(s + a)$).

For $\frac{1}{(s + 2)^3}$, we have $a = 2$ and $n = 3$.

Step 2: Apply the inverse Laplace transform formula

The general inverse Laplace transform for $\frac{1}{(s + a)^n}$ is given by:

$\mathcal{L}^{-1} \left( \frac{1}{(s + a)^n} \right) = \frac{t^{n-1}}{(n-1)!} e^{-at}$

Step 3: Substitute the values for $a = 2$ and $n = 3$

Substitute these values into the formula:

$\mathcal{L}^{-1} \left( \frac{1}{(s + 2)^3} \right) = \frac{t^{3-1}}{(3-1)!} e^{-2t}$

This simplifies to:

$\mathcal{L}^{-1} \left( \frac{1}{(s + 2)^3} \right) = \frac{t^2}{2} e^{-2t}$

Final Answer:

The inverse Laplace transform of $\frac{1}{(s+2)^3}$ is:

$\frac{t^2}{2} e^{-2t}$

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Would you like further details on the steps or any clarifications?

Here are 5 related questions to explore:

1. How do you compute the inverse Laplace transform of $\frac{1}{(s + a)^n}$ for different values of $n$?
2. What are some common Laplace transform pairs and their applications in solving differential equations?
3. How do you handle inverse Laplace transforms involving higher-order polynomials in the denominator?
4. How can the inverse Laplace transform be used to solve physical systems (like electrical circuits or mechanical systems)?
5. What is the significance of the exponential factor $e^{-at}$ in the time-domain solution?

Tip: To make the inverse Laplace transform process smoother, always check if the denominator can be factored or if it matches known standard forms for which you can directly apply the formulas.