Laplace Transform of Integral of $f(t) = t^2$

The problem requires finding the Laplace transform of the integral of a given function $f(t) = t^2$. We need to compute $ \mathcal{L} \left\{ \int_0^t f(\tau)d\tau \right\} $ where $ f(t) = t^2 $.

Laplace Transform Property for Integration

A key property of Laplace transforms simplifies this calculation. The Laplace transform of an integral is given by:

• $ \mathcal{L} \left\{ \int_0^t f(\tau)d\tau \right\} = \frac{1}{s} F(s) $, where $ F(s) $ denotes the Laplace transform of $ f(t) $, i.e., $ F(s) = \mathcal{L} \{f(t)\} $.

Calculate $F(s)$ for $f(t) = t^2$

First, determine the Laplace transform of the function $ f(t) = t^2 $. Using the standard Laplace transform formula $ \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}} $:

• For $ n=2 $, $ F(s) = \mathcal{L}\{t^2\} = \frac{2!}{s^{2+1}} = \frac{2}{s^3} $.

Calculate the Laplace Transform of the Integral

Apply the integration property using the calculated $ F(s) $:

• $ \mathcal{L} \left\{ \int_0^t \tau^2 d\tau \right\} = \frac{1}{s} F(s) $.
• Substitute $ F(s) = \frac{2}{s^3} $: $ \frac{1}{s} \times \frac{2}{s^3} $.
• Simplify the expression: $ \frac{2}{s^{1+3}} = \frac{2}{s^4} $.

Final Answer Derivation

The resulting Laplace transform of the integral of $ f(t) = t^2 $ is $ \frac{2}{s^4} $. This matches Option D.