https://file.aiquickdraw.com/mathbot/file/17267244747035599m046.jpg

Let's break down the tasks from the image:

### Part (a): Setting up the Laplace Transform integral

You are asked to find the Laplace transform of \( f(t) = t + 6 \).

The Laplace transform of a function \( f(t) \) is given by:

\[
F(s) = \mathcal{L}\{f(t)\} = \int_{0}^{\infty} f(t) e^{-st} dt
\]

For \( f(t) = t + 6 \), we substitute into the general formula:

\[
F(s) = \int_{0}^{\infty} (t + 6) e^{-st} dt
\]

This is the integral that sets up the Laplace transform for \( f(t) = t + 6 \).

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### Part (b): Finding the antiderivative

Now, we need to find the antiderivative of the expression from part (a), i.e.,

\[
\int (t + 6) e^{-st} dt
\]

This can be split into two integrals:

\[
\int t e^{-st} dt + 6 \int e^{-st} dt
\]

The integrals are:

1. **For \( \int t e^{-st} dt \)**: Use integration by parts. Let \( u = t \) and \( dv = e^{-st} dt \).
2. **For \( \int e^{-st} dt \)**: This is straightforward and equals \( \frac{e^{-st}}{-s} \).

Would you like to see the detailed steps for these integrals?

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### Relative questions:
1. How do you perform integration by parts for the first term in part (b)?
2. What is the Laplace transform of a constant function, like \( 6 \)?
3. What is the Laplace transform of \( t \)?
4. How does the Laplace transform handle linear combinations of functions?
5. Can you explain the concept of convergence for improper integrals in Laplace transforms?

### Tip:
When dealing with Laplace transforms of polynomials or linear terms like \( t + 6 \), it's useful to know standard transforms and how to use integration by parts effectively.

Set up an integral for finding the Laplace transform of f(t) = t + 6, and find the antiderivative of the given function.

This solution covers finding the Laplace transform of f(t) = t + 6, and computing its antiderivative using techniques such as integration by parts. Suitable for university-level calculus students.

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