# Who's up to the challenge? 12

Calculus Level 5

$\displaystyle \int_1^\infty \frac{\{x\} - \frac{1}{2}}{x} \, dx$

can be represented as $$\ln (\sqrt{A \pi}) - B$$, where $$A$$ and $$B$$ are positive integers. Find the value of $$2A + 2B.$$

Definition: $$\{x\} = x - \lfloor x \rfloor$$ is the fractional part of $$x$$.

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