Question

The Laplace transform of function $f(t)=t^{3}$ is:

• A. $6s^{3}$
• B. $6s^{4}$
• C. $\frac{6}{s^{3}}$
• D. $\frac{6}{s^{4}}$

Hint:

Always remember that the power of $s$ in the denominator is one greater than the power of $t$ in the time domain.

Answer 1

The Correct Option is D

Concept: The general formula for the Laplace transform of a power function $t^n$, where $n$ is a positive integer, is: \[ L\{t^n\} = \frac{n!}{s^{n+1}} \]

Step 1: Identify $n$ and apply formula.
Given $f(t) = t^3$, we have $n = 3$. \[ L\{t^3\} = \frac{3!}{s^{3+1}} \]

Step 2: Calculate the factorial and power.
- $3! = 3 \times 2 \times 1 = 6$ - $n + 1 = 3 + 1 = 4$

Step 3: Final substitution.
\[ L\{t^3\} = \frac{6}{s^4} \] This matches option (4).