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 $f(t) = t + 6$.

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?

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.