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) = $\int_0^\infty$ 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: 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.

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 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.

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