The value of $\int_{0}^{\pi / 2} \log (\operatorname{cosec} x) d x$ is

$\int_{-\pi}^\pi \frac{x \sin ^3 x}{4-\cos ^2 x} d x=$

If $\int_0^\pi \frac{x \sin x}{4 \cos^2 x + 3 \sin^2 x} dx = $

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

The value of the integral $\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x+\frac{\pi}{4}}{2-\cos 2 x} d x$ is :

$\int_0^\pi \frac{x \tan x}{\sec x+\cos x} \,d x=$