Laplace Transform of t: Formula, Proof

The Laplace transform of t is 1/s^2. In this post, we will find the Laplace transform of t by definition and by the Laplace transform formulas for integrals and derivatives.

Laplace transform of t formula

We know that the Laplace transform of integral powers of t is given by the formula:

L{tⁿ} = n!/sⁿ⁺¹.

Putting n=1, we get the Laplace transform formula of t.

L{t} = 1!/s¹⁺¹ = 1/s².

What is the Laplace transform of t?

Answer: The Laplace transform of t is 1/s².

Proof:

The definition of the Laplace transform of f(t) is given as follows:

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

Step 1: Put f(t) = t in the above definition of Laplace transform.

So the Laplace transform of t by definition is equal to

L{t} = $\int_0^\infty$ te^-st dt

Step 2: We use the integrating by parts formula, we obtain that

L{t} = $[$ t ∫e^-st dt – $\int \{\frac{d}{dt}(t) \int e^{-st} dt\} dt$ $]_0^\infty$

= $\left[ t \cdot \dfrac{e^{-st}}{-s} – \int \dfrac{e^{-st}}{-s} dt \right]_0^\infty$

= $\left[ \dfrac{te^{-st}}{-s} – \dfrac{e^{-st}}{s^2} \right]_0^\infty$

= $\lim\limits_{t \to \infty}\left[ \dfrac{te^{-st}}{-s} – \dfrac{e^{-st}}{s^2} \right]$ – (0 – 1/s²)

= 0-0 + 1/s²

= 1/s².

Thus, the Laplace transform of t is 1/s² and this is obtained by the definition of the Laplace transform.

Also Read:

Laplace transform of sin at:	a/(s2+a2)
Laplace transform of cos at:	s/(s2+a2)
Laplace transform of e-t:	1/(s+1)
Laplace transform of 1:	1/s

Find The Laplace Transform of t

We know that L{t} =1/s².

Method 1: Using the multiplication by tⁿ Laplace transform formula, we will now find the Laplace transform of t. This formula says that

L{tⁿf(t)} = (-1)ⁿ $\dfrac{d^n}{ds^n}$[F(s)], where F(s)=L{f(t)}.

Put n=1 and f(t)=1. Then F(s)=L{f(t)}=1/s.

So L{t} = (-1)¹ $\dfrac{d}{ds}$(1/s)

= – (-1/s²)

= 1/s².

Method 2: Using the Laplace transform formula for integrals, we will now find the Laplace transform of t. The formula says that

$L\{\int_0^t f(u) du \}=\frac{1}{s}$F(s), where F(s)=L{f(t)}.

Put f(u)=1. Then F(s)=L{f(u)}=1/s.

Therefore, $L\{\int_0^t du \}$ = 1/s × 1/s

⇒ L{t} = 1/s².

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