Let $\int_{-2}^{2} (\sin |x| + [x \sin x]) dx = 2(3 - \cos 2) + \beta$,where $[\cdot]$ denotes the greatest integer function. Then $\beta \sin \left(\frac{\beta}{2}\right)$ equals:

Read the following mathematical statements carefully:
$I.$ $A$ differentiable function $f$ with maximum at $x = c$ $\implies f''(c) < 0$.
$II.$ Antiderivative of a periodic function is also a periodic function.
$III.$ If $f$ has a period $T$ then for any $a \in R$,$\int\limits_0^T {f(x)\,dx} = \int\limits_0^T {f(x + a)\,dx}$.
$IV.$ If $f(x)$ has a maxima at $x = c$,then $f$ is increasing in $(c - h, c)$ and decreasing in $(c, c + h)$ as $h \to 0$ for $h > 0$. Now indicate the correct alternative.

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

The value of $\int_{-1}^{1} x^{2} e^{[x^{3}]} dx$,where $[t]$ denotes the greatest integer $\leq t$,is

The integral $\int\limits_0^{\frac{1}{2}} \frac{\ln(1 + 2x)}{1 + 4x^2} dx$ equals

The value of the integral $\int \limits_1^3 \left((x-2)^4 \sin^3(x-2) + (x-2)^{2019} + 1\right) dx$ is