Symmetrized Version of the Markovecchio Integral in the Theory of Diophantine Approximations

A new integral construction unifying the idea of symmetry proposed by Salikhov in 2007 and the integral introduced by Markovecchio in 2009 is considered. The application of this construction leads, in particular, to a sharper estimate of the measure of irrationality of the number $\pi/\sqrt{3}$.