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

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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