# 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).

## FAQs

Q1: What is the Laplace transform of sin(at)/t?

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

Q2: Find the Laplace transform of sin(2t)/t.

Answer: The Laplace transform of sin(2t)/t is tan-1(2/s).

Q3: What is the Laplace transform of sin(3t)/t?

Answer: The Laplace transform of sin(3t)/t is tan-1(3/s).

Share via: