Integral $\int_0^1\sqrt{1-x^4}dx$

Question

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.

Answer 1

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