A tough integral in disguise!

Calculus Level 4

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

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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