The area of the region $\{(x, y): y \leq \pi-|x|, y \leq|x \sin x|, y \geq 0\}$ is:

Let $\int\limits_{-2}^2(|\sin x|+[x \sin x]) d x=2(3-\cos 2)+\beta$, where [ ⋅ ] is the greatest integer function. Then $\beta \sin \left(\frac{\beta}{2}\right)$ equals:

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\left(1+x+x^2\right)\left(1-y+y^2\right), y(0)=\frac{1}{2}$. Then $(2 y(1)-1)$ is equal to

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$, then $96 p$ is equal to $\_\_\_\_$ .