The Laplace transform of \( t \cos(2t) \) can be found using the formula:

\[
\mathcal{L} \{ t f(t) \} = -\frac{d}{ds} \left( \mathcal{L} \{ f(t) \} \right)
\]

where \( f(t) = \cos(2t) \) and \( \mathcal{L} \{ \cos(2t) \} = \frac{s}{s^2 + 4} \).

Now, differentiate \( \frac{s}{s^2 + 4} \) with respect to \( s \):

\[
\frac{d}{ds} \left( \frac{s}{s^2 + 4} \right) = \frac{(s^2 + 4)(1) - s(2s)}{(s^2 + 4)^2}
= \frac{s^2 + 4 - 2s^2}{(s^2 + 4)^2}
= \frac{-s^2 + 4}{(s^2 + 4)^2}
\]

Thus, the Laplace transform of \( t \cos(2t) \) is:

\[
\mathcal{L} \{ t \cos(2t) \} = -\frac{-s^2 + 4}{(s^2 + 4)^2}
= \frac{s^2 - 4}{(s^2 + 4)^2}
\]

So, the Laplace transform of \( t \cos(2t) \) is:

\[
\mathcal{L} \{ t \cos(2t) \} = \frac{s^2 - 4}{(s^2 + 4)^2}
\]

Would you like more details or further clarification?

Here are 5 related questions:
1. How do you find the Laplace transform of \( t \sin(3t) \)?
2. What is the Laplace transform of a general function \( t^n \cos(at) \)?
3. How can you apply the convolution theorem to Laplace transforms?
4. What is the inverse Laplace transform of \( \frac{s}{s^2 + a^2} \)?
5. Can you explain the shifting property in Laplace transforms?

**Tip:** Differentiation in the Laplace domain is often used to handle expressions with time multiplication, so memorizing this technique is helpful.

Learn how to find the Laplace transform of t cos(2t) using the differentiation property of Laplace transforms. This detailed solution includes key formulas and theorems, perfect for undergraduate students studying calculus or engineering mathematics.

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