# Find Laplace transform of t^3 | Laplace of t cube

The Laplace transform of t^3 is equal to 6/s^4. In this article, we will learn how to find the Laplace transform of t cube.

## Laplace Transform of t Cube Formula

The Laplace transform formula of t cube, that is, the formula of L{t3} is given by

L{t3} = 6/s4.

## What is the Laplace Transform of t3?

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

Proof:

To find the Laplace transform of t cube by the definition of Laplace transform, let us recall the definition. The Laplace transform of f(t) is defined by

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

Step 1: Put f(t) = t3.

Therefore,

L{t3} = $\int_0^\infty$ t3 e-st dt. …(I)

Step 2: We use the theory of the Gamma function Γ(x) = $\int_0^\infty$ tx-1 e-t dx. Assume that

z=st

∴ dz=s dt ⇒ dt = dz/s. Also, t=z/s.

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

L{t3} = $\int_0^\infty \Big(\dfrac{z}{s} \Big)^3 e^{-z} \dfrac{dz}{s}$

= (1/s3+1) $\int_0^\infty z^{3+1-1} e^{-z} dz$

= (1/s4) $\Gamma(3+1)$, by the definition of the Gamma function.

= (1/s4) × 3! as we know that Γ(n+1) = n!

= 3!/s4

= 6/s4.

Therefore, the Laplace transform of t^3 is equal to 6/s4 and this is proved by the definition of Laplace transforms.

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

Laplace Transform of eat: The Laplace transform of eat is 1/(s-a).

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

Laplace transform of sin(at): The Laplace transform of sin(at) is a/(s2+a2).

Laplace transform of cos(at): The Laplace transform of cos(at) is s/(s2+a2).

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 cube?

Answer: The Laplace transform of t cube is equal to 6/s4.

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