To find the inverse Laplace transform of the given function:

\[
\mathcal{L}^{-1} \left( \frac{s}{\left(s^2 + 1\right)^2} \right)
\]

We can use the known Laplace transform pairs and the convolution theorem. 

### Step 1: Recognize a known Laplace pair
The function \(\frac{s}{s^2 + 1}\) has the known inverse Laplace transform:

\[
\mathcal{L}^{-1} \left( \frac{s}{s^2 + 1} \right) = \cos(t)
\]

### Step 2: Use the convolution theorem
We notice that the given function can be seen as a derivative with respect to \(s\) of a known transform. However, it's simpler to recognize that:

\[
\frac{s}{\left(s^2 + 1\right)^2} = \frac{1}{2} \cdot \mathcal{L}\left[t \sin(t)\right]
\]

This result follows from a standard table of Laplace transforms, where:

\[
\mathcal{L} \left[t \sin(at)\right] = \frac{2as}{(s^2 + a^2)^2}
\]

For \(a = 1\), we have:

\[
\mathcal{L} \left[t \sin(t)\right] = \frac{2s}{(s^2 + 1)^2}
\]

So, dividing by 2:

\[
\frac{s}{(s^2 + 1)^2} = \frac{1}{2} \mathcal{L} \left[t \sin(t)\right]
\]

### Step 3: Apply the inverse Laplace transform
Thus, the inverse Laplace transform is:

\[
\mathcal{L}^{-1} \left( \frac{s}{\left(s^2 + 1\right)^2} \right) = \frac{1}{2} t \sin(t)
\]

So, the final answer is:

\[
\mathcal{L}^{-1} \left( \frac{s}{\left(s^2 + 1\right)^2} \right) = \frac{1}{2} t \sin(t)
\]

Would you like further details or have any questions about this? Here are some related problems you might consider:

1. Find the inverse Laplace transform of \(\frac{s}{\left(s^2 + 4\right)^2}\).
2. Compute the inverse Laplace transform of \(\frac{1}{\left(s^2 + 1\right)^2}\).
3. Determine \(\mathcal{L}^{-1} \left( \frac{s}{\left(s^2 + 9\right)^2} \right)\).
4. Apply the convolution theorem to \(\mathcal{L}^{-1} \left( \frac{1}{\left(s^2 + 1\right)^2} \right)\).
5. Find the inverse Laplace transform of \(\frac{2s}{\left(s^2 + 4\right)^2}\).

**Tip:** When dealing with Laplace transforms involving derivatives or squared terms, it's often helpful to check standard transform tables or use the convolution theorem.

Learn how to find the inverse Laplace transform of s / ((s^2 + 1)^2) using known Laplace transform pairs and the convolution theorem. Detailed step-by-step solution provided.

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