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Homework Help: Laplace transform of te^t

  1. Feb 8, 2012 #1
    1. The problem statement, all variables and given/known data

    f(t)=te^t, find laplace

    2. Relevant equations

    3. The attempt at a solution

    I started doing integration by parts and after doing it three times I wasn't sure if I was going in the right direction/making any progress. I'm not supposed to use a table to solve this (I have to do the integral out) so could anyone give me a hint as to how to start?

    I can manipulate the original equation to
    (0 to inf for all integrals)
    ∫ te^(t(1-s)) dt and then i set u=t, du=dt, dv=e^(t(1-s)), v=e^(t(1-s))/(1-s)
    and then I end up with a longer expression and I need to integrate by parts again. is this is right direction?
  2. jcsd
  3. Feb 8, 2012 #2
    It's the right direction. After all, your t-factor in the integrand goes away, so you just have to integrate e^(t*something).
  4. Feb 8, 2012 #3
    I think I'm missing a step.

    after doing interation by parts the first time, i get

    te^[t(1-s)] / (1-s) - ∫ e^[t(1-s)] / (1-s) dt

    =te^[t(1-s)] / (1-s) - e^[t(1-s)] / (1-s)^2

    i then need to evaluate fro 1 to inf but what does that do to te^[t(1-s)], assuming s>1? t is inf and e^[t(1-s)] becomes 0?

    EDIT: nevermind, I had a brainfart, I got the answer
    Last edited: Feb 8, 2012
  5. Feb 8, 2012 #4
    Laplace transform generally has a region of convergence to the right of some axis parallel to the y-axis on the complex s plane, in this case this axis happens to be x=1.
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