https://file.aiquickdraw.com/mathbot/file/1726927815697v5pvunoh.jpg

Let's analyze the Laplace transform problem in the image. The expression provided is the inverse Laplace transform:

\[
L^{-1}\left\{ \frac{1}{(s^2 + 9)(s^2 + 4)} \right\}
\]

We need to match the correct inverse transform to the options listed below:

### Step-by-Step Solution:
1. The given Laplace expression involves the product of two quadratic terms: \( (s^2 + 9) \) and \( (s^2 + 4) \).
   
2. We recognize that:
   - \( s^2 + 9 \) corresponds to \( \sin(3t) \) in the time domain because \( \sin(3t) \) has a Laplace transform of \( \frac{3}{s^2 + 9} \).
   - \( s^2 + 4 \) corresponds to \( \sin(2t) \) because \( \sin(2t) \) has a Laplace transform of \( \frac{2}{s^2 + 4} \).

3. To find the inverse Laplace transform of the product of these two terms, we use convolution. Specifically, the inverse transform of a product is the convolution of the inverse transforms of each term:
   \[
   L^{-1} \left\{ \frac{1}{(s^2 + 9)(s^2 + 4)} \right\} = \int_0^t \sin(3\tau) \sin(2(t - \tau)) \, d\tau
   \]

4. Evaluating this integral results in:
   \[
   \frac{1}{5} \left( \sin(2t) - \sin(3t) \right)
   \]

5. Looking at the options, the only one that has a constant multiplier of \( 5 \) and fits this expression is option **(d)**:
   \[
   5\left[ \frac{\sin(3t)}{3} - \frac{\sin(2t)}{2} \right]
   \]

Thus, the correct answer is **(d)**.

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Would you like a detailed breakdown of the convolution step or need help with another part? Here are five related questions to expand your understanding:

1. How do you perform convolution in the time domain?
2. What are the Laplace transforms of other common trigonometric functions?
3. Can you explain the relationship between the Laplace and inverse Laplace transforms?
4. How do you approach finding inverse transforms of more complex expressions?
5. How does the scaling property affect the Laplace transform?

**Tip:** For products of simple Laplace-transformed functions, convolution is key when returning to the time domain.

Find the inverse Laplace transform of the expression \( \frac{1}{(s^2 + 9)(s^2 + 4)} \) and select the correct option.

This page explains how to find the inverse Laplace transform of \( \frac{1}{(s^2 + 9)(s^2 + 4)} \), using convolution and key formulas. It demonstrates step-by-step the relationship between the Laplace transform and trigonometric functions, suitable for undergraduate-level math students.

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