Laplace Transform of sint/t | Laplace of sint/t

The fraction sin(t)/t is a function with numerator sin(t) and denominator t. The Laplace transform of sin(t)/t is tan^-1(1/s). In this article, we will learn how to find the Laplace transform of sin(t)/t.

Laplace Transform of sint/t Formula

sint/t Laplace formula: The Laplace transform formula of sin(t)/t is given below:

L{sin(t)/t} = tan^-1(1/s).

What is the Laplace Transform of sint/t?

Answer: The Laplace transform of sin(t)/t is tan^-1(1/s).

Proof:

We will use the division by t Laplace transform formula here. The formula is given below.

$L\{\frac{f(t)}{t} \} =\int_s^\infty F(s) ds$, where $L\{f(t)\}=F(s)$ …(I)

Step 1: Put f(t)=sin t.

∴ F(s) = L{f(t)} = L{sin t} = 1/(s²+1)

Step 2: So from (I), we get the Laplace transform of sin(t)/t which is

L{sin(t)/t} = $\int_s^\infty \dfrac{1}{s^2+1} ds$

= $\Big[ \tan^{-1} s\Big]_s^\infty$

= tan^-1 ∞ – tan^-1 s

= π/2 – tan^-1 s

= cot^-1 s

= tan^-1 (1/s).

So the Laplace transform of sin(t)/t is tan^-1(1/s).

Find the Laplace transform of sin(t)/t. Summary: L{sin(t)/t} = tan-1(1/s).

Also Read:

Laplace transform of t:	1/s2
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

FAQs