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Integration by parts

  1. Jan 6, 2008 #1


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    [SOLVED] Integration by parts

    1. The problem statement, all variables and given/known data
    I've been starring at this for 30 minutes and can't figure out what's wrong. I end up with 1/xi instead of xi. The book says,

    [tex]\int_{-\infty}^{+\infty}te^{-t^2/2}\sin(\xi t)dt=\int_{-\infty}^{+\infty}e^{-t^2/2}\xi \cos(\xi t)dt[/tex]

    3. The attempt at a solution

    I set [itex]u=te^{-t^2/2}[/itex] and [itex]dv = \sin(\xi t)[/itex]. So I get [itex]du=e^{-t^2/2}-t^2e^{-t^2/2}[/itex] and unless I'm completely crazy, [itex]v=-\xi^{-1}\cos(\xi t)[/tex], so that

    [tex]\int_{-\infty}^{+\infty}te^{-t^2/2}\sin(\xi t)dt=te^{-t^2/2}\xi^{-1}\cos(\xi t)|_{-\infty}^{+\infty} + \int_{-\infty}^{+\infty}e^{-t^2/2}\xi^{-1}\cos(\xi t)dt - \int_{-\infty}^{+\infty}t^2e^{-t^2/2}\xi^{-1}\cos(\xi t)dt = \int_{-\infty}^{+\infty}e^{-t^2/2}\xi^{-1}\cos(\xi t)dt[/tex]

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  3. Jan 6, 2008 #2
    What is [itex] v[/itex]?

    [tex]I=-\int_{-\infty}^{+\infty}(e^{-t^2/2})'\sin(\xi t)dt=-te^{-t^2/2}\sin(\xi t)|_{-\infty}^{+\infty} + \int_{-\infty}^{+\infty}e^{-t^2/2}(\sin(\xi t))'dt = \int_{-\infty}^{+\infty}e^{-t^2/2}\xi\cos(\xi t)dt[/tex]
  4. Jan 6, 2008 #3


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    I can't entirely follow what you're doing with the u and dv. I learned partial integration as
    [tex]\int_a^b f'(x) g(x) dx = - \left. f(x) g(x) \right|_a^b + \int_a^b f(x) g'(x) dx[/tex]
    Applying this to
    [tex]f'(t) = t e^{-t^2/2}, \qquad \implies \qquad f(t) = - e^{-t^2/2}[/tex]
    [tex]g(t) = \sin(\xi t), \qquad \implies \qquad g'(t) = \xi \cos(\xi t)[/tex]
    then gives me
    [tex]\int_{-\infty}^{+\infty}te^{-t^2/2}\sin(\xi t)dt =
    \left. e^{-t^2/2} \sin(\xi t) \right|_{-\infty}^{+\infty} +
    \int_{-\infty}^{+\infty} -e^{-t^2/2}\xi \cos(\xi t)dt.
    The boundary term vanishes so that gives exactly minus the result you'd like (probably a minus error on my side) but definitely a [itex]\xi[/itex] and not [itex]\xi^{-1}[/itex]. As it should be (otherwise you'd get strange results for [itex]\xi = 0[/itex]).

    [edit]I should learn to type LaTeX even faster, Rainbow Child beat me by 5 whole minutes :smile:[/edit]
    Last edited: Jan 6, 2008
  5. Jan 6, 2008 #4


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    I see my mistake: the last term does not vanish. Thx!
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