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The question asks for the Laplace transform of \( \cosh^5 t \). To find this, we need to use properties and identities related to hyperbolic functions and the Laplace transform.

The hyperbolic cosine function raised to a power can be expressed using the binomial theorem or through identities. However, the simplest approach in this case, considering the complexity of the power, is to use the identity for the hyperbolic cosine function:

\[
\cosh t = \frac{e^t + e^{-t}}{2}
\]

Raising \(\cosh t\) to the fifth power gives:

\[
\cosh^5 t = \left( \frac{e^t + e^{-t}}{2} \right)^5
\]

Expanding this using the binomial theorem:

\[
\cosh^5 t = \frac{1}{32} \left( e^{5t} + 5e^{3t} + 10e^{t} + 10e^{-t} + 5e^{-3t} + e^{-5t} \right)
\]

Now, the Laplace transform is linear, so we can take the Laplace transform of each term separately:

\[
\mathcal{L}\{\cosh^5 t\} = \frac{1}{32} \left( \mathcal{L}\{e^{5t}\} + 5 \mathcal{L}\{e^{3t}\} + 10 \mathcal{L}\{e^{t}\} + 10 \mathcal{L}\{e^{-t}\} + 5 \mathcal{L}\{e^{-3t}\} + \mathcal{L}\{e^{-5t}\} \right)
\]

The Laplace transform of \( e^{at} \) is \( \frac{1}{s-a} \), therefore:

\[
\mathcal{L}\{\cosh^5 t\} = \frac{1}{32} \left( \frac{1}{s-5} + \frac{5}{s-3} + \frac{10}{s-1} + \frac{10}{s+1} + \frac{5}{s+3} + \frac{1}{s+5} \right)
\]

This is the Laplace transform of \( \cosh^5 t \). 

If you have any questions or need further details, please let me know!

#### Related Questions:
1. How does the binomial theorem apply to the expansion of \( \cosh^5 t \)?
2. What are the properties of the Laplace transform used in this solution?
3. How can the identity for \( \cosh t \) be derived from exponential functions?
4. What other methods can be used to compute the Laplace transform of powers of hyperbolic functions?
5. Could we use a similar approach for computing the Laplace transform of \( \sinh^5 t \)?

#### Tip:
When dealing with complex powers of hyperbolic or trigonometric functions, expressing the function in terms of exponentials can simplify the process significantly, especially when computing integrals or transforms.

Find the Laplace transform of \( \cosh^5 t \).

Learn how to compute the Laplace transform of \( \cosh^5 t \) by expanding the hyperbolic cosine function using binomial expansion and applying the linearity of the Laplace transform. This solution includes key formulas such as the Laplace transform of exponential functions.

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