The Laplace inverse of \(\tan^{-1}(s)\) can be found using standard Laplace inversion techniques and known transforms. Let's denote the inverse Laplace transform as \(\mathcal{L}^{-1}\). We aim to find:

\[
\mathcal{L}^{-1}\left\{\tan^{-1}(s)\right\}
\]

However, the function \(\tan^{-1}(s)\) does not have a standard inverse Laplace transform table entry, so it needs to be solved by recognizing that \(\tan^{-1}(s)\) can be expressed as a series expansion or using convolution techniques. But, in most practical cases, an exact inverse transform is not typically found in standard forms.

Thus, the inverse Laplace transform of \(\tan^{-1}(s)\) does not have a simple closed-form expression.

Would you like to explore the series expansion or other advanced methods for this function's Laplace inverse?

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### Follow-up Questions:
1. What is the Laplace transform of \(\tan^{-1}(t)\)?
2. How do we compute inverse Laplace transforms using convolution?
3. What are some common functions with known inverse Laplace transforms?
4. Can you use partial fraction decomposition to simplify inverse Laplace transforms?
5. What is the significance of the series expansion of a function in Laplace transforms?

**Tip:** When a direct Laplace inverse is not easily available, exploring a function's series expansion or convolution can sometimes offer alternative approaches.

Explore the process of finding the Laplace inverse of tan inverse(s). Learn about key concepts such as Laplace transforms, series expansion, and convolution techniques. This advanced calculus problem requires knowledge of inverse Laplace transform methods and theorems.

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