We are tasked with finding the Laplace transform of \((t \cos(2t))^2\). Let's denote this function as \( f(t) = (t \cos(2t))^2 \).

### Step 1: Simplify the Expression

We first simplify \( f(t) = (t \cos(2t))^2 \):

\[
f(t) = t^2 \cos^2(2t)
\]

Using the trigonometric identity:

\[
\cos^2(2t) = \frac{1 + \cos(4t)}{2}
\]

We can now rewrite \( f(t) \) as:

\[
f(t) = t^2 \cdot \frac{1 + \cos(4t)}{2} = \frac{t^2}{2} + \frac{t^2 \cos(4t)}{2}
\]

### Step 2: Apply the Laplace Transform

Now, apply the Laplace transform to both terms separately.

#### First Term: \( \frac{t^2}{2} \)

The Laplace transform of \( t^n \) is:

\[
\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}
\]

So, for \( \frac{t^2}{2} \):

\[
\mathcal{L}\left\{\frac{t^2}{2}\right\} = \frac{1}{2} \cdot \frac{2!}{s^{2+1}} = \frac{1}{s^3}
\]

#### Second Term: \( \frac{t^2 \cos(4t)}{2} \)

The Laplace transform of \( t^n \cos(at) \) is:

\[
\mathcal{L}\{t^n \cos(at)\} = \frac{s^2 - a^2}{(s^2 + a^2)^{n+1}}
\]

For \( t^2 \cos(4t) \), with \( n = 2 \) and \( a = 4 \):

\[
\mathcal{L}\left\{\frac{t^2 \cos(4t)}{2}\right\} = \frac{1}{2} \cdot \frac{s^2 - 16}{(s^2 + 16)^3}
\]

### Step 3: Combine the Results

Finally, we add the Laplace transforms of the two terms:

\[
\mathcal{L}\{(t \cos(2t))^2\} = \frac{1}{s^3} + \frac{1}{2} \cdot \frac{s^2 - 16}{(s^2 + 16)^3}
\]

This is the Laplace transform of \( (t \cos(2t))^2 \).

Would you like further details or clarifications on any of the steps?

### Relative Questions:
1. What is the Laplace transform of \( t^3 \sin(at) \)?
2. How do you derive the Laplace transform of \( \cos(at) \)?
3. Can you explain the convolution theorem in Laplace transforms?
4. What is the inverse Laplace transform of \( \frac{1}{s^3} \)?
5. How does the shifting theorem apply to Laplace transforms?

### Tip:
When dealing with products of \( t \) and trigonometric functions, it's useful to break them down into simpler terms using trigonometric identities before applying the Laplace transform.

Explore how to find the Laplace transform of (tcos(2t))^2 with a detailed breakdown. Learn about key concepts such as Laplace transforms of polynomials and trigonometric functions. This solution involves advanced formulas and identities, making it suitable for university-level math students.

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