The value of $\int_{-2}^{2}\left|3 x^{2}-3 x-6\right| d x$ is ...... .

Let $T > 0$ be a fixed number. Suppose $f$ is a continuous function such that for all $x \in \mathbb{R}, f(x + T) = f(x)$. If $I = \int_{0}^{T} f(x) dx$,then the value of $\int_{3}^{3 + 3T} f(2x) dx$ is

$\int\limits_0^\pi (x \cdot \sin^2 x \cdot \cos x) dx =$

$\int_{-2}^{\pi} \frac{\sin^2 x}{[\frac{x}{\pi}] + \frac{1}{2}} \,dx$ is equal to (where $[\cdot]$ denotes the greatest integer function).

The value of the integral $\frac{48}{\pi^{4}} \int_{0}^{\pi} \left(\frac{3 \pi x^{2}}{2} - x^{3}\right) \frac{\sin x}{1 + \cos^{2} x} dx$ is equal to

$\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\cot x}} = $ . . . . . . .