All Roads Lead To π\pi

Calculus Level 3

01(lnx)2(1x)2dx \displaystyle \large \int_{0} ^{1} \dfrac{(\ln x)^2}{(1-x)^{2}} \, dx

If the value of the integral above equals πa!a+1\dfrac{\pi ^{a!}}{a+1}, where aa is a positive integer, find aa.

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