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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