If \( \displaystyle \int_{\pi/6}^{\pi/3} \sqrt{\,1-\sin 2x\,}\,dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3} \), where \( \alpha, \beta \) and \( \gamma \) are rational numbers, then \( 3\alpha + 4\beta - \gamma \) is equal to _________.
