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:

The value of $\int\limits_{0}^{20\pi} (\sin^4 x + \cos^4 x) dx$ is equal to:

Let [.] denote the greatest integer function. Then the value of $\int\limits_{0}^{3} \left( \frac{e^{x} + e^{-x}}{[x]!} \right) dx$ is :

Let [.] denote the greatest integer function. Then

$$ \int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{12(3+[x])}{3+\left[\sin x\right]+\left[\cos x\right]} \right) dx $$
is equal to :