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Integral with sin(x), cos(2x), and sqrt

  1. Jan 12, 2008 #1

    gop

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

    [tex]\int_{0}^{\pi/4}\frac{\sin x}{\sqrt{\cos2x}}dx[/tex]

    2. Relevant equations



    3. The attempt at a solution

    I used the standard substitutions (like y = tan(x/2)) and the substitution to get rid of the square root). I also tried to rewrite cos(2x) in many different ways but in the end I always end up with more square roots and/or power terms of which I can't compute the integral.
     
    Last edited: Jan 12, 2008
  2. jcsd
  3. Jan 12, 2008 #2

    dynamicsolo

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    Maybe you ought to show something you tried: perhaps there's just an algebra error somewhere?

    You wrote cos 2x as 2(cos x)^2 - 1 , used u = cos x , and that didn't work?

    [EDIT: Whoa-hey! I just spotted something else: this is a Type II improper integral. The denominator is zero at the upper limit of the integration. I think the indefinite integral works OK, but you'll want to use a limit at the (pi)/4 end to see whether this thing actually converges or not...]
     
    Last edited: Jan 12, 2008
  4. Jan 12, 2008 #3

    Gib Z

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    No need to use a limit (unless your doing this in the formal sense), the value just plugs straight in. For the value of the indefinite integral at pi/4, i got 0.
     
  5. Jan 12, 2008 #4

    dynamicsolo

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    As for the behavior at the vertical asymptote, I agree that the limit turns out to be unnecessary; however, I do not agree that the value at pi/4 is zero... (This improper integral does converge all right.)
     
  6. Jan 12, 2008 #5
    @dynamicsolo I don't think he has to consider any limits, with your substitution the indefinite integral reads

    [tex]I=-\frac{\sqrt{2}}{2}\,\ln\left(\sqrt{2}\,\cos x+\sqrt{\cos(2\,x)}\right)[/tex]

    which is well defined in [itex][0,\frac{\pi}{4}][/itex]
     
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