The Laplace transform of e^t sint is equal to 1/[(s-1)^{2}+1]. Note that e^{t} 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 e^{t} sint is denoted by L{e^{t} sint}. The Laplace transform formula of e^{t} sint is given below:

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

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## Find the Laplace of e^{t} sint

Answer: The Laplace of e^{t} sint is 1/[(s-1)^{2}+1]. |

**Explanation:**

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{e^{at} f(t)} = F(s-a) **…(I)**

Now, to find the Laplace of e^{t} 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/(s^{2}+a^{2}). Thus, from formula (I), we get that

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

Therefore, the Laplace transform of e^{t}sint is equal to 1/[(s-1)^{2}+1].

**Main Article:**

Laplace Transform: Definition, Table, Formulas, Properties

Laplace Transform Problems:

- Laplace Transform of sint/t
- Find L{sin
^{2}t} - Find L{t cost}
- Find Laplace of (1-cost)/t
- Laplace Transform of (1-e
^{t})/t - Find L{sinh at}
- Find L{cosh at}

## FAQs

**What is the Laplace transform of e**

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

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.