# Laplace Transform of t^n: Formula, Proof

The function tn is the nth 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.

## Laplace Transform of tn Formula

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

L{tn} = n!/sn+1.

## What is the Laplace Transform of tn?

Answer: The Laplace transform of tn is n!/sn+1.

Proof:

We will find the Laplace transform of tn 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) = tn.

So the Laplace transform of tn by definition is equal to

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

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

z=st

∴ dz=s dt ⇒ dt = dz/s. Also, t=z/s.

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

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

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

= (1/sn+1) $\Gamma(n+1)$, by the definition of the Gamma function.

= (1/sn+1) × n! as we know that Γ(n) = (n-1)!

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

Thus, the Laplace transform of tn is n!/sn+1.

Important Notes:

• tn Laplace transform: L{tn} = n!/sn+1.
• t2 Laplace transform: L{t2} = 2/s3.
• t3 Laplace transform: L{t3} = 6/s4.
• t4 Laplace transform: L{t4} = 24/s5.

## FAQs

Q1: What is the Laplace transform of t2?

Answer: The Laplace transform of t2 is 2/s3.

Q2: What is the Laplace transform of t3?

Answer: The Laplace transform of t3 is 6/s4.

Q3: What is the Laplace transform of t4?

Answer: The Laplace transform of t4 is 24/s5.

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