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Math Help - Evaluate Integral

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    Evaluate Integral

    Evaluate Integral

    \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

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

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

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

    \varphi  = \int_0^{\pi /2} {\frac{{\sin ^n x}}<br />
{{\cos ^n x + \sin ^n x}}\,dx}  = \int_0^{\pi /2} {\frac{{\cos ^n x}}<br />
{{\cos ^n x + \sin ^n x}}\,dx} .

    This yields 2\varphi  = \frac{\pi }<br />
{2} and the rest follows.
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