To find the Laplace transform of \(\cosh^5(t)\), we will first express \(\cosh^5(t)\) in terms of basic hyperbolic functions using power reduction formulas or binomial expansions.

### Step 1: Express \(\cosh^5(t)\)

Recall the identity for \(\cosh(t)\):

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

Thus, \(\cosh^5(t)\) can be written as:

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

We can now expand this using the binomial theorem. The binomial expansion for \((a + b)^5\) is:

\[
(a + b)^5 = \sum_{k=0}^{5} \binom{5}{k} a^{5-k} b^k
\]

Here, \(a = e^t\) and \(b = e^{-t}\). Expanding the expression:

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

### Step 2: Take the Laplace transform

The Laplace transform of \(e^{at}\) is \(\frac{1}{s - a}\), provided \(s > a\). Using this, we can compute the Laplace transform of each term:

- \( \mathcal{L}\{e^{5t}\} = \frac{1}{s - 5} \)
- \( \mathcal{L}\{e^{3t}\} = \frac{1}{s - 3} \)
- \( \mathcal{L}\{e^{t}\} = \frac{1}{s - 1} \)
- \( \mathcal{L}\{e^{-t}\} = \frac{1}{s + 1} \)
- \( \mathcal{L}\{e^{-3t}\} = \frac{1}{s + 3} \)
- \( \mathcal{L}\{e^{-5t}\} = \frac{1}{s + 5} \)

Thus, the Laplace transform of \(\cosh^5(t)\) becomes:

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

### Final Expression:

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

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

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Would you like further clarification on any of the steps?

Here are some related questions you might find helpful:
1. What is the Laplace transform of \(\sinh(t)\)?
2. How do you apply the binomial theorem to complex expressions?
3. What is the Laplace transform of \(\cosh(t)\)?
4. How can we find the inverse Laplace transform of a function?
5. How does the Laplace transform simplify the analysis of differential equations?

**Tip:** When dealing with hyperbolic functions, their properties are very similar to trigonometric functions, so many techniques for transforming trigonometric functions can be adapted to hyperbolic ones.

Explore how to compute the Laplace transform of cosh^5(t) using the binomial expansion and properties of Laplace transforms. This detailed solution involves expanding hyperbolic functions and applying Laplace transform formulas.

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