Laplace Transform of 1 | Laplace Transform of a Constant

The Laplace transform of 1 is 1/s and the Laplace transform of a constant is constant × 1/s. We know that the Laplace transform is a method of transforming a time variable function into a complex variable function. In this article, we will find the Laplace transform of 1.

What is the Laplace Transform of 1?

Answer: The Laplace transform of 1 is 1/s.

Proof:

Recall, the Laplace transform of a function f(t), denoted by L{f(t)} or F(s), is given by the following formula:

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

Putting f(t)=1 in the above formula, we get the Laplace transform of 1 which is

L{1} = $\int_0^\infty$ 1 ⋅ e^-st dt

$=\int_0^\infty$ e^-st dt

Take 1 as the first function and e^-st as the second function. Then integrating by parts, we obtain that

L{1} = $=\int_0^\infty$ e^-st dt

$=\left[\dfrac{e^{-st}}{-s}- \int [\dfrac{d}{dt}(1) \cdot \int e^{-st} dt] \right]_0^\infty$

$=\left[\dfrac{e^{-st}}{-s}- 0 \right]_0^\infty$

$=\lim\limits_{t \to \infty}\dfrac{e^{-st}}{-s}- \dfrac{e^{-s\cdot 0}}{-s}$

$=0- \dfrac{1}{-s}$

$=\dfrac{1}{s}$.

Thus, we have proven that the Laplace transform of 1 is 1/s.

Also Read: Inverse Laplace Transform of 1

Find the Laplace transform of 1. Summary: The Laplace transform of 1 is 1/s.

Laplace Transform of Constants

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

Proof:

In the above definition (I) of Laplace transforms, we put f(t)=c. Then the Laplace transform of a constant c by definition is given as follows:

L{c} = $\int_0^\infty$ c ⋅ e^-st dt

= c $\int_0^\infty$ e^-st dt

= c L{1}

= c ⋅ 1/s as the Laplace transform of 1 is 1/s

= c/s

Thus, we have proven that the Laplace transform of a constant is equal to constant times 1/s.

Find the Laplace transform of c, where c is a constant. Summary: The Laplace transform of c is c/s.

Question: Find the Laplace transform of 2.

Solution:

We know that the Laplace transform of c is c/s, that is L{c}=c/s. Putting c=2, we obtain that the Laplace transform of 2 is 2/s.

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