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Deriving some Laplace transforms

  1. May 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the Laplace transform of each of the following functions:

    ....

    2. f(t)=eatcos(bt)

    3. f(t)=tn, were n is a positive integer

    2. Relevant equations

    As you well know, taking the Laplace of f(t) means ∫f(t)e-stdt from 0 to ∞

    3. The attempt at a solution

    These problems are tripping me up, since integration by parts goes on forever.

    ∫eatcos(bt)e-stdt =
    ∫et(a-s)cos(bt)dt =

    I suppose I should call et(a-s) "dv" (?)
    ----> v = t(a-s)/(a-s)
    ----> u = cos(bt)
    ----> du = -bsin(bt) dt

    Still, I don't see where this gets me. Help, please!
     
  2. jcsd
  3. May 16, 2010 #2
    Do the integration by parts again, and then collect the integral you are interested in on one side.

    This is like the following simpler integral:

    [tex] \int e^x \sin x dx [/tex]
    [tex] \int e^x \sin x dx = -e^x \cos x - \int -e^x \cos x dx [/tex] (one IBP)
    [tex] \int e^x \sin x dx = -e^x \cos x + e^x \sin x - \int e^x \sin x dx [/tex] (second IBP)
    [tex] 2 \int e^x \sin x dx = e^x \sin x - e^x \cos x [/tex] (collect like terms)
    etc...
     
  4. May 16, 2010 #3
    In the case of this particular problem, what integral am I interested in?
     
  5. May 16, 2010 #4
    Well, you want

    [tex] \int_{0}^{\infty} e^{t(a-s)} \cos (bt) dt [/tex]

    which is related, but with, you know, constants and things.
     
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