To find the Laplace transform of \(\frac{\sin(\omega t)}{t}\), we can use the following known result from Laplace transform tables:

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

So, the Laplace transform of \(\frac{\sin(\omega t)}{t}\) is:

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

### Do you want further details or have any questions? 

### Related Questions:
1. What is the Laplace transform of \(\sin(\omega t)\)?
2. How does the Laplace transform change if \(\omega\) is replaced by a function of \(t\)?
3. What is the inverse Laplace transform of \(\tan^{-1}\left(\frac{\omega}{s}\right)\)?
4. How do we apply Laplace transforms to solve differential equations?
5. What is the significance of the Laplace transform in control systems?

### Tip:
When solving Laplace transform problems, always check if the function can be simplified or if a known Laplace transform formula can be directly applied.

Explore the Laplace transform of sin(ωt)/t with a detailed explanation and formula. Learn how to apply Laplace transforms to solve differential equations and understand its significance in control systems. Suitable for advanced mathematics enthusiasts.

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