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Math Help - A nice integral

  1. #1
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    A nice integral

    It looks very nice but ... it's quite simple

     \int_0^{\pi} \frac{ dx}{ 1 + \sin^{\cos(x)}(x)}
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  2. #2
    MHF Contributor red_dog's Avatar
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    I=\int_0^{\pi}\frac{dx}{1+(\sin x)^{\cos x}}

    Let x=\pi-t\Rightarrow dx=-dt

    x=0\Rightarrow t=\pi, \ x=\pi\Rightarrow t=0

    Then I=-\int_{\pi}^0\frac{dt}{1+(\sin(\pi-t))^{\cos(\pi-t)}}=\int_0^{\pi}\frac{dt}{1+(\sin t)^{-\cos t}}=

    =\int_0^{\pi}\frac{(\sin t)^{\cos t}}{1+(\sin t)^{\cos t}}dt=\int_0^{\pi}\left(1-\frac{1}{1+(\sin t)^{\cos t}}\right)dt=

    =\left. t\right|_0^{\pi}-I=\pi-I\Rightarrow I=\frac{\pi}{2}
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