Definitely not a semicircle

Calculus Level 3

1(1x)1xxx1-(1-x)^{1-x}x^x The curve looks like a semicircle with a radius 0.5.

Which is larger,011(1x)1xxxdx\displaystyle\int_0^1 1-(1-x)^{1-x}x^x dx or π8\dfrac{\pi}8?

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