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Math Help - How to calculate this integral?

  1. #1
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    How to calculate this integral?

    Hey, how are you? I was troubled by calculating f(a)=\int_0^\pi \log(1+a \cos x)dx. Here, |a|<1 is real.

    I do know we can differentiating under the integral, however, I could not see any more...
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  2. #2
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    Re: How to calculate this integral?

    That seems like a reasonable approach. Differentiate with respect to a and exchange the derivative and integral. You should get a rational function of cos x which you can integrate. Then you will have to integrate with respect to a and evaluate the constant by looking at a=0.

    So what do you get for the derivative of log(1+a cos x) with respect to a?

    - Hollywood
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  3. #3
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    Re: How to calculate this integral?

    In fact, I do the series expansion. It is difficult to integrate the derivative of log(1+a cos x) with respect to a, isn't it?
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  4. #4
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    Re: How to calculate this integral?

    Yes, I think that would qualify as a difficult integral. But it's not impossible. You can substitute u=\tan\frac{x}{2} to convert any rational combination of trig functions to a rational function of u, which can then be integrated using partial fractions. But then it might not be possible to integrate the result with respect to a.

    I like your method.

    - Hollywood
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