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:

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

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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.