# Integration of fractional part and floor function

Calculus Level 4

Evaluate the following definite integral :

$\huge\int\limits_{0}^{1}\left\{ (-1)^{\left\lfloor \frac{1}{x} \right\rfloor} \frac{1}{x} \right\}\,\mathrm dx$

where $$\{x\}$$ denotes the fractional part of $$x$$ and $$\left\lfloor x\right\rfloor$$ denotes the greatest integer function ( floor function ).

Note: Use the following definition to solve the problem.

$\left\{ x\right\} = x - \left\lfloor x\right\rfloor~\forall~x\in\Bbb R$

where $$\Bbb R$$ denotes the set of all reals.