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It looks like the image contains a Laplace transform problem involving the function \( e^{-2t} \int_0^t \frac{\cos(2t) \, dt}{t} \).

Here is a breakdown of the steps:

### Given:
You are asked to find the Laplace transform of:

\[
L \left\{ e^{-2t} \int_0^t \frac{\cos(2t) \, dt}{t} \right\}
\]

### Step 1: Use known transforms
The integral suggests that the cosine function \( \cos(2t) \) is involved, and we know the Laplace transform of \( \cos(at) \) is:

\[
L \left\{ \cos(at) \right\} = \frac{s}{s^2 + a^2}
\]

In your notes, it looks like the transform for \( \cos(2t) \) was written as:

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

### Step 2: Solve the Laplace transform involving exponentials
The Laplace transform of \( e^{-2t} f(t) \) involves shifting the variable \( s \) by 2. So:

\[
L \left\{ e^{-2t} f(t) \right\} = F(s + 2)
\]

This would apply to the Laplace transform of the entire expression, where the exponential function modifies the result by a shift in \( s \).

### Step 3: Evaluate the final result
Now, applying the above steps would involve transforming the integral and handling the shift by 2 due to \( e^{-2t} \).

---

This provides an overview of the process based on your notes. Would you like more details on the integral solution or clarification on handling Laplace transforms with shifts? 

Here are some follow-up questions you might want to explore:
1. What is the general process for handling Laplace transforms of products like \( e^{-at} f(t) \)?
2. How does the shifting theorem apply to different functions like sine and cosine?
3. Can you use convolution properties for functions involving integrals?
4. How do you interpret the result when the function inside the integral involves division by \( t \)?
5. What would happen if the limits of the integral change?

**Tip:** For Laplace transforms of shifted functions \( e^{-at} f(t) \), always remember to adjust the \( s \)-domain by shifting by \( a \).

Find the Laplace transform of e^(-2t) * integral(0 to t) [cos(2t) dt / t].

This problem focuses on finding the Laplace transform of the function e^(-2t) * integral(0 to t) [cos(2t) dt / t]. It uses the Laplace transform of cosine and the shifting theorem to solve. Suitable for advanced calculus or engineering students.

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