Laplace Transform of e^t sint | L{e^t sint}

The Laplace transform of e^t sint is equal to 1/[(s-1)2+1]. Note that et sint is a product of an exponential function and a sine function. Here we learn how to find the Laplace of e^t sint.

The Laplace Transform of et sint is denoted by L{et sint}. The Laplace transform formula of et sint is given below:

L{et sint} = $\dfrac{1}{(s-1)^2+1}$.

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Find the Laplace of et sint

Answer: The Laplace of et sint is 1/[(s-1)2+1].


Let f(t) be a function with Laplace transform F(s) = L{f(t)}. Here we will use the following property of Laplace transforms:

L{eat f(t)} = F(s-a) …(I)

Now, to find the Laplace of et sint, in the above formula we put

a=1, f(t) =sint.

So F(s) = L{sin t} = $\dfrac{1}{s^2+1}$

as we know L{sin at} = a/(s2+a2). Thus, from formula (I), we get that

L{et sint} = F(s-1) = $\dfrac{1}{(s-1)^2+1}$ as F(s)= $\dfrac{1}{s^2+1}$

Therefore, the Laplace transform of etsint is equal to 1/[(s-1)2+1].

Main Article: 

Laplace Transform: Definition, Table, Formulas, Properties

Laplace Transform Problems:


What is the Laplace transform of et sint?

Answer: The Laplace transform of et sint is 1/[(s-1)2+1], that is, L{et sint} = 1/[(s-1)2+1].

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