Find the Laplace transform of  .

To find the Laplace transform of \( \frac{\cos(\sqrt{t})}{\sqrt{t}} \), we can use the definition of the Laplace transform:

\[ \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt \]

where \( s \) is a complex number.

First, let's rewrite the given function as \( f(t) = \frac{\cos(\sqrt{t})}{\sqrt{t}} \). Then, we can substitute \( f(t) \) into the Laplace transform integral:

\[ F(s) = \mathcal{L}\left\{\frac{\cos(\sqrt{t})}{\sqrt{t}}\right\} = \int_0^\infty e^{-st} \frac{\cos(\sqrt{t})}{\sqrt{t}} \, dt \]

This integral might be a bit tricky to evaluate directly. We can simplify it by using a property of the Laplace transform, which states that if \( \mathcal{L}\{f(t)\} = F(s) ______ _____ ________ ___ ___ __________ __________ _________.
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