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How to integrate a sinc function?

  1. Sep 8, 2011 #1
    Can anyone show me how to do this:

    [tex]\int_a^b \left[\frac {\sin x}{x}\right]^2 dx [/tex]
     
  2. jcsd
  3. Sep 8, 2011 #2
    It depends on you purpose, but in general the best bet may be power series methods.

    The Taylor series for sin2 is

    [itex]\sin^2 x = -\sum^\infty_{k=1} \frac{(-1)^{k} 2^{2k-1}x^{2k}}{(2k)!}[/itex]

    So we have

    [itex]\frac{\sin^2 x}{x^2} = -\sum^\infty_{k=1} \frac{(-1)^{k} 2^{2k-1}x^{2k-2}}{(2k)!}[/itex]

    Then you can try term-wise integration and the fundamental theorem of calculus to get a general form for the answer as a power series.
     
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