# A very fickle function

Calculus Level 4

The graph of $$f(x) = \sin ( \ln x )$$ (as shown above) looks innocent enough to noticeably oscillate as $$x$$ increases. However, as $$x$$ approaches $$0$$, the oscillations grow rapidly, making $$f(x + \epsilon)$$ vary greatly from $$f(x)$$ around this region, even at very infinitesimal values of $$\epsilon$$.

That said, $$f(x)$$ will cross the $$x$$-axis for an infinite number of times from $$x=0$$ to $$x=1$$, creating several regions of the first quadrant enclosed by the curve and the $$x$$-axis.

If the sum of these regions is $$A$$, then determine $$\big\lfloor 10^5 A \big\rfloor$$.

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