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} = $\dfrac{1}{(s-1)^2+1}$.
Table of Contents
Find the Laplace of et sint
| Answer: The Laplace of et 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{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:
- Laplace Transform of sint/t
- Find L{sin2t}
- Find L{t cost}
- Find Laplace of (1-cost)/t
- Laplace Transform of (1-et)/t
- Find L{sinh at}
- Find L{cosh at}
FAQs
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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