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Integral of sin(x^2) help

  1. Jul 29, 2009 #1
    hi everyone can someone please help me out. This is not homework just getting ready for school
    integral of sin(x^2)
     
  2. jcsd
  3. Jul 29, 2009 #2

    Cyosis

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    Are you familiar with complex integration and the residue theorem?
     
  4. Jul 29, 2009 #3
    no i am not
     
  5. Jul 29, 2009 #4

    Hootenanny

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    In that case, I would try a substitution.
     
  6. Jul 29, 2009 #5

    Cyosis

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    Seeing as you say you're getting ready for school, are you still in high school? Also I just noticed that you didn't specify an interval to integrate over. Did you just make up this integral yourself? The reason I am asking this is that this function does not have a primitive function in terms of elementary functions.
     
  7. Jul 29, 2009 #6

    Hootenanny

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    Well noted Cyosis, I presumed that by 'school' the OP meant grad school, which looking back now may have not been a wise assumption.
     
  8. Jul 29, 2009 #7
    [itex]\int \sin(x^2)\,dx[/itex] is not elementary.

    So "hints" like "try substitution" are not helpful.
     
  9. Jul 29, 2009 #8

    Hootenanny

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    Indeed it is, as has already been pointed out.
    Really? How about substituting u=x2, then expanding sin(u) about u=0 and performing term-wise integration? Does this not give the power-series definition of the Fresnel function S(x)?
     
  10. Jul 29, 2009 #9
    I think your substitution hint implied either u-substitution or integration by parts. There is no need to make a substitution to expand sin(x2) out into its power series.
     
  11. Oct 28, 2009 #10
    it is a fresnel intetgral
     
  12. Oct 28, 2009 #11
    There is no systematic way to compute Fresnel integrals as I know.
    But there are several approximation methods

    I found Peter L. Volegov's code in Matlab central. It uses a method proposed in the following : (ith an error of less then 1x10-9)

    Klaus D. Mielenz, Computation of Fresnel Integrals. II
    J. Res. Natl. Inst. Stand. Technol. 105, 589 (2000), pp 589-590

    Or simply wiki Fresnel Integrals
     
  13. Oct 28, 2009 #12
    by the way it is suprising that nobody above heard of Fresnels.
     
  14. Oct 28, 2009 #13

    It is useful if you want to derive an asymptotic expression for the case of the integral from zero to R for large R.
     
  15. Oct 28, 2009 #14
    If you're integrating from 0 to R, then for small R, you simply integrate the Taylor expansion term by term.

    If R is large, you write the integral as an integral from zero to infinity minus the integral from R to infinity. The former integral is is number which you ca easily obtaoin using contour integration methods. The latter you compute by doing the substitution x^2 = u as suggested by Hootenanny, and then you do a relpeated partial integration, where you integrate the sin and differentiate the 1/sqrt(u). You iterate this, each time integrating the trigonometric term and differentiating the 1/u^(n+1/2). This then yields an asymptotic expansion with the last unevaluated integral as an error term.
     
    Last edited: Oct 28, 2009
  16. Oct 28, 2009 #15

    lurflurf

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    There is no systematic way to compute sine as I know.
    But there are several approximation methods
    thus it would be quite a surprize if fresnel integrals were easier
     
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