A very fickle function

Calculus Level 4

The graph of f(x)=sin(lnx) f(x) = \sin ( \ln x ) (as shown above) looks innocent enough to noticeably oscillate as xx increases. However, as xx approaches 00, the oscillations grow rapidly, making f(x+ϵ) f(x + \epsilon) vary greatly from f(x) f(x) around this region, even at very infinitesimal values of ϵ \epsilon .

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

If the sum of these regions is AA, then determine 105A \big\lfloor 10^5 A \big\rfloor .

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