The Laplace transform of t^2, i.e the Laplace of t square is equal to 2/s^{3}. In this article, we will learn how to find the Laplace transform of t^{2}.

The Laplace transform t^{2} (t square) is denoted by L{t^{2}}, and its formula is given as follows:

L{t^{2}} = 2/s^{3}.

This follows from the Laplace formula of t^{n}: L{t^{n}} = n!/s^{n+1} by putting n=2.

**Main Article:** Laplace Transform: Definition, Table, Formulas, Properties & Examples

Table of Contents

## What is the Laplace Transform of t^{2}?

**Answer:** The Laplace transform of t^{2} is equal to 2/s^{3}.

*Proof:*

The Laplace transform of f(t) by definition is given by

L{f(t)} = $\int_0^\infty$ f(t) e^{-st} dt.

So to find the Laplace transform of t^{2} by definition, we need to follow below steps.

**Step 1:** Put f(t) = t^{2}.

Therefore,

L{t^{2}} = $\int_0^\infty$ t^{2} e^{-st} dt.** …(I)**

**Step 2:** In order to compute the above integral, let us use the theory of the Gamma function Γ(x) = $\int_0^\infty$ t^{x-1} e^{-t} dx. Assume that

z=st

∴ dz=s dt ⇒ dt = dz/s. Also, t=z/s.

t | z |

0 | 0 |

∞ | ∞ |

**Step 3:** The equation **(I)** then implies that

L{t^{2}} = $\int_0^\infty \Big(\dfrac{z}{s} \Big)^2 e^{-z} \dfrac{dz}{s}$

= (1/s^{2+1}) $\int_0^\infty z^{2+1-1} e^{-z} dz$

= (1/s^{3}) $\Gamma(2+1)$, by the definition of the Gamma function.

= (1/s^{3}) × 2! as we know that Γ(n+1) = n!

= 2!/s^{3}

= 2/s^{3}.

Therefore, the Laplace transform of t^2 is equal to 2/s^{3} and this is proved by the definition of Laplace transforms.

**Read Also:**

Laplace Transform of e^{at}: | The Laplace transform of e^{at} is 1/(s-a). |

Laplace transform of sin(at): | The Laplace transform of sin(at) is a/(s^{2}+a^{2}). |

Laplace transform of cos(at): | The Laplace transform of cos(at) is s/(s^{2}+a^{2}). |

Laplace transform of a constant: | The Laplace transform of a constant c is c/s. |

Inverse Laplace transform of constant: | The inverse Laplace transform of c is cδ(t), where δ(t) is the Dirac delta function. |

## FAQs

**Q1: What is the Laplace transform of t square?**

Answer: The Laplace transform of t^{2} is L{t^{2}} = 2/s^{3}.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.