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To find the Laplace transform \( F(s) = \mathcal{L} \{ f(t) \} \) of the function \( f(t) = 7 - 5t^4 + \sin(7t) \), we use the linearity of the Laplace transform and the known transforms of each term separately.

The function is:
\[
f(t) = 7 - 5t^4 + \sin(7t)
\]

### Step-by-Step Solution

1. **Laplace Transform of Constant Term \( 7 \):**
   \[
   \mathcal{L}\{7\} = \frac{7}{s}
   \]

2. **Laplace Transform of \( -5t^4 \):**
   The Laplace transform of \( t^n \) is given by \( \mathcal{L}\{t^n\} = \frac{\Gamma(n+1)}{s^{n+1}} \), where \( \Gamma(n+1) = n! \).
   
   For \( -5t^4 \):
   \[
   \mathcal{L}\{-5t^4\} = -5 \cdot \frac{4!}{s^{5}} = -5 \cdot \frac{24}{s^{5}} = -\frac{120}{s^5}
   \]

3. **Laplace Transform of \( \sin(7t) \):**
   The Laplace transform of \( \sin(at) \) is \( \mathcal{L}\{\sin(at)\} = \frac{a}{s^2 + a^2} \).
   
   For \( \sin(7t) \):
   \[
   \mathcal{L}\{\sin(7t)\} = \frac{7}{s^2 + 49}
   \]

### Combining Results

Using the linearity property:
\[
F(s) = \frac{7}{s} - \frac{120}{s^5} + \frac{7}{s^2 + 49}
\]

### Final Answer

\[
F(s) = \frac{7}{s} - \frac{120}{s^5} + \frac{7}{s^2 + 49}
\]

Would you like further details or have any questions?

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**Related Questions:**

1. How do you find the Laplace transform of a general polynomial function?
2. What is the Laplace transform of other trigonometric functions like \( \cos(at) \)?
3. How do Laplace transforms help in solving differential equations?
4. What are some common applications of the Laplace transform in engineering?
5. How does the Laplace transform differ from the Fourier transform?

**Tip:** Remember, the Laplace transform is particularly useful for solving differential equations because it turns differentiation into multiplication by \( s \), simplifying the process significantly.

Find the Laplace transform F(s) = L{f(t)} of the function f(t) = 7 - 5t^4 + sin(7t).

Step-by-step solution to finding the Laplace transform of f(t) = 7 - 5t^4 + sin(7t). This example demonstrates the use of Laplace transforms for polynomial and trigonometric functions, and includes key formulas and linearity properties. Suitable for undergraduate students studying differential equations.

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