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To find the inverse Laplace transform of the function

\[
F(s) = \frac{5s + 9}{s^2 + 49},
\]

we can break it down using known Laplace transform pairs and linearity.

1. **Separate Terms**: Notice that we can rewrite \( F(s) \) as follows:
   \[
   F(s) = \frac{5s}{s^2 + 49} + \frac{9}{s^2 + 49}.
   \]

2. **Use Known Laplace Transforms**:
   - The term \(\frac{5s}{s^2 + 49}\) corresponds to the Laplace transform of a cosine function. Specifically,
     \[
     \mathcal{L}^{-1} \left( \frac{5s}{s^2 + 49} \right) = 5 \cos(7t).
     \]
   - The term \(\frac{9}{s^2 + 49}\) corresponds to the Laplace transform of a sine function. Specifically,
     \[
     \mathcal{L}^{-1} \left( \frac{9}{s^2 + 49} \right) = \frac{9}{7} \sin(7t).
     \]

3. **Combine Results**:
   \[
   f(t) = 5 \cos(7t) + \frac{9}{7} \sin(7t).
   \]

Thus, the inverse Laplace transform is:
\[
f(t) = 5 \cos(7t) + \frac{9}{7} \sin(7t).
\]

Would you like further details or have any questions about this solution?

Here are some related questions to explore further:
1. How can we find the inverse Laplace transform of functions with higher-order polynomials?
2. What are the typical forms of inverse Laplace transforms for functions involving trigonometric expressions?
3. How does the linearity property of the Laplace transform simplify inverse calculations?
4. Can we use partial fraction decomposition for similar Laplace transform problems?
5. How do we apply the shifting theorem in the context of inverse Laplace transforms?

**Tip:** Always try to identify parts of the function that resemble known Laplace transform pairs for easier calculation.

Find the inverse Laplace transform \( f(t) = \mathcal{L}^{-1} \{ F(s) \} \) of the function \( F(s) = \frac{5s + 9}{s^2 + 49} \).

Learn how to find the inverse Laplace transform of the function F(s) = (5s + 9)/(s^2 + 49) using trigonometric transforms. This solution involves breaking down the function into parts and applying known Laplace transform pairs for sine and cosine functions. Suitable for undergraduate mathematics students.

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