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To solve this Laplace transform problem, let's break it down step-by-step.

### Problem Analysis
We are asked to find the Laplace transform of the expression:
\[
\mathcal{L} \left\{ 6 + u_1(t)e^{-4t} \cos(\pi t) \right\}
\]
where:
- \( 6 \) is a constant.
- \( u_1(t) \) represents the unit step function at \( t = 1 \).
- \( e^{-4t} \cos(\pi t) \) is a damped cosine function modulated by the unit step function \( u_1(t) \).

### Step 1: Laplace Transform of Each Term Separately
1. **Constant Term \(6\):**
   \[
   \mathcal{L} \{6\} = \frac{6}{s}
   \]

2. **Term with Unit Step and Damped Cosine:**
   To find the Laplace transform of \( u_1(t)e^{-4t} \cos(\pi t) \), we use the **time-shifting property** of the Laplace transform:
   \[
   \mathcal{L} \{ u_a(t) f(t - a) \} = e^{-as} \mathcal{L} \{ f(t) \} \Big|_{t \to t - a}
   \]
   Here, \( a = 1 \), and we need the Laplace transform of \( e^{-4t} \cos(\pi t) \).

### Step 2: Laplace Transform of \( e^{-4t} \cos(\pi t) \)
For \( e^{-4t} \cos(\pi t) \), we use the standard formula:
\[
\mathcal{L} \{ e^{at} \cos(bt) \} = \frac{s - a}{(s - a)^2 + b^2}
\]
In our case:
- \( a = -4 \)
- \( b = \pi \)

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

### Step 3: Apply the Time-Shift for \( u_1(t) \)
Using the time-shifting property with \( u_1(t) \), we get:
\[
\mathcal{L} \{ u_1(t) e^{-4(t - 1)} \cos(\pi (t - 1)) \} = e^{-s} \cdot \frac{s + 4}{(s + 4)^2 + \pi^2}
\]

### Step 4: Combine Results
The complete Laplace transform is:
\[
\mathcal{L} \left\{ 6 + u_1(t)e^{-4t} \cos(\pi t) \right\} = \frac{6}{s} + e^{-s} \cdot \frac{s + 4}{(s + 4)^2 + \pi^2}
\]

### Final Answer
\[
\mathcal{L} \left\{ 6 + u_1(t)e^{-4t} \cos(\pi t) \right\} = \frac{6}{s} + \frac{e^{-s} (s + 4)}{(s + 4)^2 + \pi^2}
\]

Would you like further details on any step, or have any questions?

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### Additional Practice Questions

1. Compute the Laplace transform of \( u_2(t)e^{-3t} \sin(2t) \).
2. Determine the inverse Laplace transform of \( \frac{e^{-2s} (s + 3)}{s^2 + 6s + 10} \).
3. Find the Laplace transform of \( t u_3(t) e^{-t} \).
4. Apply the time-shifting property to \( u_1(t)(t^2 + 5t) \).
5. Calculate \( \mathcal{L} \left\{ 7 \cos(4t) \right\} \).

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### Tip
When using the Laplace transform with unit step functions, always apply the time-shifting property carefully to handle any delays in the function accurately.

Compute the Laplace transform of \(6 + u_1(t)e^{-4t} \cos(\pi t)\).

Learn to compute the Laplace transform of \(6 + u_1(t)e^{-4t} \cos(\pi t)\), covering the unit step function, exponential decay, cosine function, and the time-shifting property. Step-by-step solution with relevant formulas included.

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