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The image you provided contains a set of Laplace transform problems, along with some integrals and inverse Laplace transforms. Let's tackle the problems systematically, starting with the first few:

### Problem Analysis:
1. **Find \(\mathcal{L} \left\{ e^{t}\left(\cos(2t) + \frac{1}{2} \sinh(2t)\right) \right\}\)**

   - To solve this, use the linearity of the Laplace transform and the known transforms for \( \cos(at) \) and \( \sinh(at) \).

2. **Evaluate \(\mathcal{L} \left\{ \int_{0}^{u} \frac{\sin(u)}{u} \, du \right\}\)**

   - This involves applying the Laplace transform to an integral function. Use the convolution theorem for Laplace transforms here.

3. **Find \(\mathcal{L} \left\{ t e^{-t} \sin(2t) \right\}\)**

   - This can be solved using the shift theorem for the Laplace transform, as well as the standard formula for \(\sin(bt)\).

4. **Find \(\mathcal{L} \left\{ t^2 e^{-3t} \right\}\)**

   - Use the property \(\mathcal{L} \{ t^n e^{-at} \} = \frac{n!}{(s+a)^{n+1}}\) to solve this.

5. **Evaluate \(\int_{0}^{\infty} t e^{-3t} \sin(t) \, dt\)**

   - This is an improper integral and can be solved using integration by parts or by recognizing it as a Laplace transform evaluated at a specific point.

Would you like to proceed with detailed solutions for these or focus on any specific problem? 

### Here are 5 related questions:
1. How does the convolution theorem simplify solving Laplace transforms involving integrals?
2. What is the Laplace transform of \( \cos(at) \) and how can it be used in problem 1?
3. How do shifting properties of Laplace transforms help with \( e^{-at} \) terms?
4. What is the inverse Laplace transform and how does it differ from the direct transform?
5. How can partial fractions be used to solve inverse Laplace transform problems like question 13?

### Tip:
For Laplace transforms involving products like \( t^n f(t) \), remember to use the differentiation property: \(\mathcal{L}\{t^n f(t)\} = (-1)^n \frac{d^n}{ds^n} \mathcal{L}\{f(t)\}\).

Solve the Laplace transform and inverse Laplace transform problems listed, which include finding Laplace transforms of functions, evaluating integrals, and using Dirac-delta functions.

Explore a comprehensive problem set on Laplace and inverse Laplace transforms, including calculations with Dirac-delta and unit step functions. Ideal for students at the undergraduate level, this set covers key mathematical concepts and theorems for solving Laplace transforms.

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