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Math Help - Another interesting integral

  1. #1
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    Another interesting integral

    evaluate the integral  \int_{0}^{\frac{\pi}{2}} x \cot(x) ~dx

    In fact , it is not a challenge problem for an experienced integrator
    but i believe that this may help remind us of some techniques
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  2. #2
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    Quote Originally Posted by simplependulum View Post
    evaluate the integral  \int_{0}^{\frac{\pi}{2}} x \cot(x) ~dx

    In fact , it is not a challenge problem for an experienced integrator
    but i believe that this may help remind us of some techniques
    let x=u and \cot x \ dx = dv. then integration by parts gives us: I=\int_0^{\frac{\pi}{2}} x \cot x \ dx = -\int_0^{\frac{\pi}{2}} \ln(\sin x) \ dx. thus:
    -I=\frac{\pi}{2} \ln 2 + \int_0^{\frac{\pi}{2}} \ln(\sin(x/2)) \ dx + \int_0^{\frac{\pi}{2}} \ln(\cos(x/2)) \ dx

    =\frac{\pi}{2} \ln 2 + \int_0^{\frac{\pi}{2}} \ln(\sin(x/2)) \ dx + \int_{\frac{\pi}{2}}^{\pi} \ln(\sin(x/2)) \ dx

    =\frac{\pi}{2} \ln 2 + \int_0^{\pi} \ln(\sin(x/2)) \ dx=\frac{\pi}{2} \ln 2 + 2\int_0^{\frac{\pi}{2}} \ln(\sin x) \ dx

    =\frac{\pi}{2} \ln 2 - 2I. therefore: I=\frac{\pi}{2}\ln 2.
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  3. #3
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    terrific !
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