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Laplace Transform question

  1. Jul 20, 2011 #1
    1. The problem statement, all variables and given/known data

    Laplace Transform of e-t sin t

    2. Relevant equations

    f2a613fc61132e4b8f053ed85030a651.png

    3. The attempt at a solution

    I have the solution, but I am unable to figure out how the denominator becomes 1/[(s + 1)2 + 1]
     
  2. jcsd
  3. Jul 20, 2011 #2

    rock.freak667

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    Homework Helper

    The presence of the eat would case the shift from 's' to 's-a'. This is why it is called the shift theorem, it's mainly used in the inverse laplace transform.

    So you know that L{sint} = 1/(s2+1)

    and following shift theorem L(eatsint) = 1/[(s-a)2+1].

    You can derive it too using the integral formula.
     
  4. Jul 20, 2011 #3
    I am unable to derive it from the integral formula. I need to see the steps. I'm fairly certain I've been able to integrate it correctly, but I keep getting a repetitive e^-t sin t or e^-t cos t when I integrate.
     
  5. Jul 20, 2011 #4
    So you're probably ending up with something like ∫e-t sin t dt on both sides of the equation; just move one to the other side and combine them as like terms. There's a good example of a similar problem on Wikipedia with ∫ex cos x dx:
    http://en.wikipedia.org/wiki/Integration_by_parts#Integrals_with_powers_of_x_or_ex
     
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