Answer: The Laplace transform of t square is equal to 2/s^{3}, that is, L{t^{2}} = 2/s^{3}. In this post, we will learn about the formula of L{t^{2}} along with its proof.

## Laplace Transform of t^{2 }Formula

The Laplace of t square formula is given as follows

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

## Laplace Transform of t^{2 }Proof

We will now prove that the Laplace transform of t^{2} is L{t^{2}} = 2/s^{3}. By the definition of Laplace ransforms, we have

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

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

Put x=st

∴ dx=s dt

⇒ dt = dx/s. Note that t=x/s.

t | x |

0 | 0 |

∞ | ∞ |

So from **(I)**, the Laplace of t^{2} will be equal to

L{t^{2}} = $\int_0^\infty (\dfrac{x}{s})^2 e^{-x} \dfrac{dx}{s}$

= $\dfrac{1}{s^3}$ $\int_0^\infty x^2 e^{-x} dx$

= $\dfrac{1}{s^3} \Gamma(3)$ where the Gamma function is defined as follows: $\Gamma(z)=\int_0^\infty x^{z-1} e^{-x} dx$

= $\dfrac{1}{s^3} \times 2$ as we know that $\Gamma(n)=(n-1)!$

= $\dfrac{2}{s^3}$

So the Laplace transform of t square by the definition is equal to 2/s^{3}.

More Laplace Transforms:

## Question-Answer

**Question 1:** Find the Laplace transform of 2t^{2}.

**Solution:**

As L{t^{2}} = 2/s^{3}, the Laplace of 2t^{2} will be equal to

L{2t^{2}} = 2L{t^{2}}⇒ L{2t ^{2}} = $2 \cdot \dfrac{2}{s^3}$⇒ L{2t ^{2}} = $\dfrac{4}{s^3}$. |

So the Laplace transform of 2t^{2} is equal to 4/s^{3}, that is, L{2t^{2}} = 4/s^{3}.

## FAQs

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

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