Arcsin and Natural Log

Calculus Level pending

Consider the functions \(f(x)=\arcsin(x)\) and \(g(x)=\ln(x)\), both for all \(0<x\leqslant{c}\). If function \(g\) strictly undergoes a positive, vertical translation, it will at some point be tangent to function \(f\) at \(x=c\). Find the area \(A\) bounded by \(f\) and the translated \(g\) that is tangent to \(f\).

Note that \(A\) can be expressed as \(\sqrt{a}-b\) where \(a\) and \(b\) are positive integers with \(a\) being square-free. Enter your answer as \(a^2+b^2\).

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