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Let's go through the problems one by one and solve them. The text contains four main tasks involving Laplace transforms and differential equations.

### Problem 1: Find the Laplace transform of the following functions

#### a) \(\frac{1 - e^{-2x}}{2x}\)

The Laplace transform of \(\frac{1 - e^{-2x}}{2x}\) can be found using the known Laplace transforms. We recognize this as the form:

\[
\mathcal{L}\left\{\frac{1 - e^{-2x}}{2x}\right\} = \mathcal{L}\left\{\frac{1}{2x}\right\} - \mathcal{L}\left\{\frac{e^{-2x}}{2x}\right\}
\]

Using properties of the Laplace transform:

\[
\mathcal{L}\left\{e^{-ax} f(x)\right\} = F(s+a)
\]

We get:

\[
\mathcal{L}\left\{\frac{1}{2x}\right\} = \frac{1}{2s}, \quad \mathcal{L}\left\{\frac{e^{-2x}}{2x}\right\} = \frac{1}{2(s+2)}
\]

So,

\[
\mathcal{L}\left\{\frac{1 - e^{-2x}}{2x}\right\} = \frac{1}{2s} - \frac{1}{2(s+2)}
\]

#### b) \(x \int_0^x e^{-xt} \cos t \, dt\)

This requires convolution theorem for Laplace transforms. Let's denote \( f(t) = x e^{-xt} \cos t \).

#### c) \(e^{-x} \int_0^x (x - t)^2 \cos^2 t \, dt\)

This requires integration and substitution to simplify before applying the Laplace transform.

### Problem 2: Find the inverse Laplace transform of the following functions

#### a) \(\cot^{-1}(s + 1)\)

#### b) \(\frac{e^{-s}}{(s^2 + 4)^2}\)

#### c) \(\frac{s - 6}{s^3 - 2s^2 + 6s}\)

### Problem 3: Evaluate the integrals using Laplace transform

#### a) \(\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x} \, dx\)

#### b) \(\int_0^\infty \frac{e^{-ax} \sin(bx)}{x} \, dx\)

### Problem 4: Solve the following differential equations using Laplace transform

#### a) 
\[ y'' + 3y' - y = x \]
\[ y(0) = 1, \, y'(0) = -1 \]

#### b) 
\[ y'' + 2y' + 5y = 3e^{-x} \sin x \]
\[ y(0) = 0, \, y'(0) = 3 \]

### Solving Problem 1a in detail:

Given:
\[ \frac{1 - e^{-2x}}{2x} \]

The Laplace transform is:
\[ \mathcal{L}\left\{\frac{1 - e^{-2x}}{2x}\right\} = \frac{1}{2s} - \frac{1}{2(s+2)} \]

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Would you like to proceed with detailed solutions for each part, or do you have any specific question or part you want to focus on? 

Here are five related questions you might want to explore:

1. How to find the Laplace transform of complex functions?
2. What are the properties of the Laplace transform?
3. How to apply the convolution theorem in Laplace transforms?
4. How to solve differential equations using the Laplace transform?
5. What are common Laplace transform pairs?

**Tip:** Always remember to check initial conditions when solving differential equations using Laplace transforms.

Explore detailed solutions for problems involving Laplace transforms, differential equations, and inverse Laplace transforms. Includes step-by-step explanations and application of key formulas and integration techniques. Suitable for advanced undergraduates and beyond.

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