\[ \displaystyle\int_{1}^{2} \dfrac{ \sqrt { \lfloor x \rfloor ^{2} + 2 \lfloor x \rfloor \lbrace x \rbrace - \lbrace x \rbrace^{2} } }{ \lfloor x \rfloor + \lbrace x \rbrace } \text{dx} \]

If the integral above equsls to \(A\). And suppose \( A\) is equal to \( (\sqrt{a} - b ) \left( b + \dfrac{ \pi}{ a\sqrt {a} } \right) \), find \( \ln(a \times b) \).

**Details and Assumption**:

\( \lfloor . \rfloor \) denotes floor function.

\( \lbrace . \rbrace \) denotes fractional part function.

\( \ln\) denotes logarithm to the base \(e\).

\(a \) and \( b \) are coprime positive integers.

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