Let ${I_1} = \int_a^{\pi - a} {xf(\sin x)dx}$ and ${I_2} = \int_a^{\pi - a} {f(\sin x)dx}$,then ${I_2}$ is equal to

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_0^{\pi /2} \log(\sin x) \, dx = $

The value of the integral $\int_{0}^{\pi / 2} \frac{1}{1+(\tan x)^{-101}} d x$ is equal to

Evaluate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x) dx$,where $f(x) = \sin |x| + \cos |x|$ for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.

If $[t]$ denotes the greatest integer $\leq t$,then the value of $\frac{3(e-1)^2}{e} \int \limits_1^2 x^2 e^{[x]+[x^3]} dx$ is: