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Thread: Evaluate Integral

  1. #1
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    Evaluate Integral

    Evaluate Integral

    $\displaystyle \int_0^\frac{\pi}{2}\frac{sin^{25} x}{\cos^{25} x+\sin^{25} x}dx$
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  2. #2
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    Quote Originally Posted by perash View Post
    Evaluate Integral

    $\displaystyle \int_0^\frac{\pi}{2}\frac{sin^{25} x}{\cos^{25} x+\sin^{25} x}dx$
    More generally:

    $\displaystyle \int_0^{\pi /2} {\frac{{\sin ^n x}}
    {{\cos ^n x + \sin ^n x}}\,dx} = \frac{\pi }
    {4}.$ (No matter what $\displaystyle n$ is.)

    Let $\displaystyle \varphi =\int_0^{\pi /2} {\frac{{\sin ^n x}}
    {{\cos ^n x + \sin ^n x}}\,dx}.$ Substitute $\displaystyle u = \frac{\pi }
    {2} - x,$

    $\displaystyle \varphi = \int_0^{\pi /2} {\frac{{\sin ^n x}}
    {{\cos ^n x + \sin ^n x}}\,dx} = \int_0^{\pi /2} {\frac{{\cos ^n x}}
    {{\cos ^n x + \sin ^n x}}\,dx} .$

    This yields $\displaystyle 2\varphi = \frac{\pi }
    {2}$ and the rest follows.
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