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Differentiation under the integral

  1. Nov 20, 2007 #1
    1. The problem statement, all variables and given/known data
    [tex] f(\alpha) = \int \log(1+ \alpha \cos(x))dx [/tex]

    I am supposed to differentiate w.r.t alpha and then integrate to find f(alpha).

    My book says that there should be a factor of pi in the answer but I do not get one. Does anyone else?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 20, 2007 #2

    quasar987

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    no bounds to that integral?
     
  4. Nov 20, 2007 #3
    Sorry. It goes from 0 to pi.
     
  5. Nov 21, 2007 #4

    arildno

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    Okay, so we differentiate:
    [tex]f'(\alpha)=\int_{0}^{\pi}\frac{\cos(x)}{1+\alpha\cos(x)}dx=\frac{1}{\alpha}\int_{0}^{\pi}(1-\frac{1}{1+\alpha\cos(x)})dx,f(0)=0[/tex]
    See if this brings you any further.

    The substitution [tex]u=\tan(\frac{x}{2})[/tex] might well be helpful.
     
  6. Nov 21, 2007 #5
    is log(1+acos(x)) = cos(x)/(1+acos(x)) ?
     
  7. Nov 21, 2007 #6

    Gib Z

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    No, he was doing what the thread title says. It's the derivative of your integrand with respect to alpha.
     
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