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Estimating the integral of a decreasing trigonometric function

  1. Nov 21, 2013 #1
    Hi,

    So this is part of an assignment for my numerical analysis class.

    The integral is this:

    [tex]

    \int_0^{\infty} e^{-x} \cos^2 (x^2) dx

    [/tex]

    We are instructed to evaluate the integral from 0 to some large A using numerical methods (which I'm fine with), and then estimate the tail, ie the integral from A to infinity.

    Basically we need to come up with some method to estimate and bound the remainder in the tail.
    My idea was to substitute [tex] u = x^2 [/tex] which gives [tex] \int_A^{\infty} \frac{e^{- \sqrt{u}} \cos^2(u)}{\sqrt{u}} du [/tex]
    and then I guess take the maximum of the cos function as 1 and just remove it, and use the remaining function, which I think would be quite easy to integrate and would give a maximum bound.
    Any thoughts greatly appreciated.

    Regards,
    Adam
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 21, 2013 #2

    Dick

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    I don't see why you just don't use cos(x^2)^2<=1 to get a bound on the tail. But sure, you can do it that way too.
     
    Last edited: Nov 21, 2013
  4. Nov 21, 2013 #3
    Yeah I just realised that my way is retarded and ends up with the exact same result.

    Thanks!
     
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