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Definitely not a semicircle

Calculus Level 3

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

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

\displaystyle\int_0^1 1-(1-x)^{1-x}x^x dx

Can't determine.

\dfrac{\pi}8

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