Integrating Floors And Ceilings

Calculus Level 2

$\large \int_{\frac{1}{8}}^{\frac{1}{2}} \left \lfloor \ln \left \lceil \dfrac{1}{x} \right \rceil \right \rfloor dx = \, ?$

 Notations: $$\lfloor \cdot \rfloor$$ denotes the floor function and $$\lceil \cdot \rceil$$ denotes the ceiling function.

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