If ∫_((π)/6)^((π)/3) √(1- sin 2 x) d x=α+β √ 2 +γ √ 3, where α, β and γ are rational numbers, then 3 α+4 β-γ is equal to ____.

If $\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2 x} d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}$, where $\displaystyle \alpha, \beta$ and $\displaystyle \gamma$ are rational numbers, then $\displaystyle 3 \alpha+4 \beta-\gamma$ is equal to $\displaystyle \_\_\_\_$.