https://file.aiquickdraw.com/mathbot/file/17304148987529y1wepr8.jpg

I need to know what to put into the boxes for the right answer. Hint: I can't put 1/s^2 in the boxes

To compute the Laplace transform of \( f(t) = t \), we use the integral:

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

This requires integration by parts:

1. **Set up for integration by parts**:
   - Let \( u = t \) and \( dv = e^{-st} \, dt \).
   - Then \( du = dt \) and \( v = -\frac{e^{-st}}{s} \).

2. **Apply integration by parts**:
   - Using the formula \( \int u \, dv = uv - \int v \, du \):
   
   \[
   \int_0^{\infty} t e^{-st} \, dt = \left[ -\frac{t e^{-st}}{s} \right]_0^{\infty} + \int_0^{\infty} \frac{e^{-st}}{s} \, dt
   \]

3. **Evaluate the boundary terms**:
   - As \( t \to \infty \), \( t e^{-st} \to 0 \).
   - At \( t = 0 \), the term is also zero.
   - This results in \( 0 - 0 = 0 \) in the first box.

4. **Simplify the remaining integral**:
   - We are left with \( \frac{1}{s} \int_0^{\infty} e^{-st} \, dt \).
   - This integral evaluates to \( \frac{1}{s^2} \).

5. **Final Answer**:
   - The Laplace transform of \( t \) is \( \frac{1}{s^2} \).

---

Here’s what goes in each box:

1. First box: **0**
2. Second box: **0**
3. Third box: **1**
4. Fourth box: **0**
5. Fifth box: **0**

Would you like more details, or do you have any questions about this solution?

### Related Questions
1. How do you perform integration by parts in general?
2. What is the significance of the Laplace transform in engineering?
3. Why does \( t e^{-st} \) go to zero as \( t \to \infty \)?
4. Can the Laplace transform be applied to other types of functions?
5. How would the Laplace transform differ for a function like \( f(t) = t^2 \)?

### Tip
For functions involving \( t \) multiplied by exponentials, integration by parts is typically a good approach.

Learn how to compute the Laplace Transform of f(t) = t by applying integration by parts. This detailed solution shows each step, including boundary evaluations and integral simplifications. Ideal for students in undergraduate calculus and differential equations.

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