A very fickle function

Calculus Level 5

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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