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I am asked to show $\int_0^1\sqrt{1-x^4}dx=\frac{\{\Gamma(1/4)\}^2}{6\sqrt{2\pi}}$. I know the gamma function is defined by $\Gamma(n)=\int_0^\infty x^{n-1}e^{-x}dx$. I tried to substituted $x^2=\sin(t)$ but couldn't go further. I am really questioned how a radical function can convert to an exponential one? :-0 Thank you.

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up vote 9 down vote accepted

HINT: Change variables $t=x^4$, and use Euler's integral of the first kind to express the answer as beta function.

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Excellent! Thanks for the hint. –  Nancy Rutkowskie Sep 22 '12 at 15:40
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