Let $f:R \to R$ and $g:R \to R$ be continuous functions,then the value of the integral $\int_{-\pi/2}^{\pi/2} [f(x) + f(-x)][g(x) - g(-x)] \, dx$ is:

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

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_{2 - \log 3}^{3 + \log 3} \frac{\log (4 + x)}{\log (4 + x) + \log (9 - x)} \, dx = $

The value of the integral $\int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x$ is

The value of $\int_1^3 \sqrt{3 + x^3} \,dx$ lies in the interval