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How to integrate cos( sin(x) ) from 0 to pi

  1. Apr 17, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]
    \int_{0}^{\pi} \cos ( \sin x ) \mbox{d}x
    [/tex]




    3. The attempt at a solution

    If I use [itex]u = \pi-x [/itex] I get :



    [tex]
    \int_{0}^{\pi} \cos ( \sin x ) \mbox{d}x = \int_{0}^{\pi} \sin ( \cos x ) \mbox{d}x
    [/tex]

    but then what?
     
  2. jcsd
  3. Apr 17, 2009 #2

    Dick

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    Re: integral

    That reminds me of a definition of the Bessel function in terms of integrals...
     
  4. Apr 17, 2009 #3
    Re: integral

    So n = 0 here:

    [tex] J_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \cos (n \tau - x \sin \tau) \,\mathrm{d}\tau. [/tex]
     
  5. Apr 17, 2009 #4

    Dick

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    Re: integral

    Sure. n=0, x=1.
     
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