\[\large \int_{1}^{2} \left(x^{\lfloor x^2 \rfloor} + {\lfloor x^2 \rfloor}^{x} \right) \, dx\]

The integral above can be expressed as\[ \dfrac{a}{b} + \sqrt{c} + \dfrac{\sqrt{d}}{e} + \dfrac{2^{\sqrt{f}}-2^{\sqrt{g}}}{\ln h } + \dfrac{i - 3^{\sqrt{j}}}{ \ln k}.\] What is the sum of all of these constants from \(\displaystyle{a}\) to \(\displaystyle{k}?\)

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