To determine the Laplace Transform of $f(t) = 5t^3$ using the definition and integration by parts, we proceed as follows:

Definition of the Laplace Transform:

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

Here, $f(t) = 5t^3$. Substituting into the definition: $F(s) = \int_0^\infty e^{-st} \cdot 5t^3 \, dt$

Factor out the constant $5$: $F(s) = 5 \int_0^\infty t^3 e^{-st} \, dt$

We now compute the integral using integration by parts.

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Integration by Parts Formula:

$\int u \, dv = uv - \int v \, du$

Let: $u = t^3 \quad \text{and} \quad dv = e^{-st} \, dt$

Step 1: Compute $du$ and $v$

$u = t^3 \implies du = 3t^2 \, dt$ $dv = e^{-st} \, dt \implies v = \frac{-e^{-st}}{s}$

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Step 2: Apply the Integration by Parts Formula

$\int t^3 e^{-st} \, dt = uv - \int v \, du$ Substitute $u$, $v$, $du$, and $dv$: $\int t^3 e^{-st} \, dt = \left( t^3 \cdot \frac{-e^{-st}}{s} \right) \Big|_0^\infty - \int_0^\infty \frac{-e^{-st}}{s} \cdot 3t^2 \, dt$

Step 3: Evaluate Boundary Terms

The term $\left( t^3 \cdot \frac{-e^{-st}}{s} \right) \Big|_0^\infty$ evaluates as follows:

• At $t = \infty$: $e^{-st} \to 0$ (exponentially decays), so the term vanishes.
• At $t = 0$: $t^3 \cdot e^{-st} = 0$.

Thus, the boundary term is $0$.

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Step 4: Simplify Remaining Integral

$\int t^3 e^{-st} \, dt = -\frac{3}{s} \int_0^\infty t^2 e^{-st} \, dt$

Now repeat integration by parts for $\int t^2 e^{-st} \, dt$.

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General Pattern

Repeating the process for $t^2$, $t^1$, and $t^0$, we derive the Laplace Transform of $t^n$: $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$

For $f(t) = 5t^3$, $n = 3$: $\mathcal{L}\{5t^3\} = 5 \cdot \frac{3!}{s^{3+1}} = 5 \cdot \frac{6}{s^4} = \frac{30}{s^4}$

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Final Answer:

$\mathcal{L}\{5t^3\} = \frac{30}{s^4}$

Would you like a more detailed explanation or examples of similar problems?

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5 Related Questions:

1. What is the Laplace Transform of $t^n$ for arbitrary $n$?
2. How does the choice of $u$ and $dv$ in integration by parts affect the solution?
3. Can the Laplace Transform handle piecewise functions like the unit step function?
4. What is the Laplace Transform of $f(t) = e^{at}t^n$?
5. How is the Laplace Transform used in solving differential equations?

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Tip:

For functions of the form $t^n e^{at}$, the Laplace Transform is best derived using the formula: $\mathcal{L}\{t^n e^{at}\} = \frac{n!}{(s-a)^{n+1}}$