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The given equation represents the Laplace transform of a function \( f(t) \), which is written as:

\[
F(s) = \mathcal{L}\{f(t)\} = \int_0^{\infty} (t + 6) e^{-st} \, dt
\]

We are tasked with finding the antiderivative corresponding to this equation, with the constant term equal to 0.

### Step 1: Break down the integral

The expression involves integrating \( (t + 6) e^{-st} \) from \( t = 0 \) to \( t = \infty \). We can split this into two integrals:

\[
F(s) = \int_0^{\infty} t e^{-st} \, dt + \int_0^{\infty} 6 e^{-st} \, dt
\]

### Step 2: Solve each integral

#### Integral 1: \( \int_0^{\infty} t e^{-st} \, dt \)
This is a standard Laplace transform, and it has a known result:

\[
\int_0^{\infty} t e^{-st} \, dt = \frac{1}{s^2}
\]

#### Integral 2: \( \int_0^{\infty} 6 e^{-st} \, dt \)
This is another standard Laplace transform, and it also has a known result:

\[
\int_0^{\infty} e^{-st} \, dt = \frac{1}{s}
\]

Multiplying by 6 gives:

\[
\int_0^{\infty} 6 e^{-st} \, dt = \frac{6}{s}
\]

### Step 3: Combine the results

Now, summing both results, we get:

\[
F(s) = \frac{1}{s^2} + \frac{6}{s}
\]

This is the Laplace transform of the function, which we have found through direct integration. This represents the antiderivative with the constant term 0.

Would you like more details on any part of the process?

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Here are some related questions:
1. How do you compute Laplace transforms of other functions like \( t^n e^{-st} \)?
2. How does the shifting property of Laplace transforms work?
3. How does this result relate to inverse Laplace transforms?
4. How do different initial conditions affect Laplace transforms in general?
5. What is the role of the \( s \)-domain in control theory?

**Tip:** When solving Laplace integrals, it's useful to recall standard forms of basic integrals, like those for \( e^{-st} \) and \( t e^{-st} \), as they commonly appear in differential equations and engineering problems.

Find the antiderivative (with constant term 0) corresponding to the Laplace transform of the function F(s) = L{f(t)} = integral from A to B of (t + 6)e^(-st) dt, where A = 0 and B = infinity.

Explore the detailed solution to finding the antiderivative corresponding to the Laplace transform of the function (t + 6)e^(-st), integrated from 0 to infinity. This advanced calculus problem involves key concepts like the Laplace Transform and integral calculus.

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