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

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{2-\sin \theta}{2+\sin \theta}\right) d \theta$ is equal to

The value of $\int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$ is

If $f(x) = \cos(\tan^{-1}x)$,then the value of the integral $\int_{0}^{1} x f''(x) dx$ is

$\int_0^{2 \pi} \sin ^6 x \cos ^5 x \, dx$ is equal to

Let $f:[-1, 2] \rightarrow [0, \infty)$ be a continuous function such that $f(x) = f(1-x)$ for all $x \in [-1, 2]$. Let $R_1 = \int_{-1}^2 x f(x) dx$,and $R_2$ be the area of the region bounded by $y = f(x)$,$x = -1$,$x = 2$,and the $x$-axis. Then