The function t^{n} is the n^{th} power of t. The Laplace transform of t^n is equal to n!/s^{n+1}. In this article, we will learn how to find the Laplace transform of real powers of t.

Table of Contents

## Laplace Transform of t^{n} Formula

The formula of the Laplace transform of t^{n}, denoted by L{t^{n}}, is given below:

L{t^{n}} = n!/s^{n+1}.

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

**Answer:** The Laplace transform of t^{n} is n!/s^{n+1}.

*Proof:*

We will find the Laplace transform of t^{n} by the definition of Laplace transform. By definition, the Laplace transform of f(t) is given the following integral formula:

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

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

So the Laplace transform of t^{n} by definition is equal to

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

**Step 2:** To use the theory of the Gamma function Γ(x) = $\int_0^\infty$ t^{x-1} e^{-t} dx, let us assume that

z=st

∴ dz=s dt ⇒ dt = dz/s. Also, t=z/s.

t | z |

0 | 0 |

∞ | ∞ |

**Step 3:** Therefore, from **(I)** we get that

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

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

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

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

= $\dfrac{n!}{s^{n+1}}$.

Thus, the Laplace transform of t^{n} is n!/s^{n+1}.

Find the Laplace transform of t^n.Summary:The Laplace transform of t^n is n!/s ^{n+1}. |

**Important Notes:**

- t
^{n }Laplace transform: L{t^{n}} = n!/s^{n+1}. - t
^{2 }Laplace transform: L{t^{2}} = 2/s^{3}. - t
^{3 }Laplace transform: L{t^{3}} = 6/s^{4}. - t
^{4 }Laplace transform: L{t^{4}} = 24/s^{5}.

**Read Also:**

Concept of Laplace Transform: Definition, Table, Formulas, Properties & Examples |

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 constant: The Laplace transform of 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**

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

**Q2: What is the Laplace transform of t**

^{3}?Answer: The Laplace transform of t^{3} is 6/s^{4}.

**Q3: What is the Laplace transform of t**

^{4}?Answer: The Laplace transform of t^{4} is 24/s^{5}.