Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous functions. Then the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} [f(x)+f(-x)][g(x)-g(-x)] \, dx$ is

If ${I_n} = \int_0^{\pi /4} {{\tan ^n}\theta \,d\theta }$,then ${I_8} + {I_6}$ equals

Which of the following statements are true?

If $I = \frac{2}{\pi} \int_{-\pi / 4}^{\pi / 4} \frac{dx}{(1 + e^{\sin x})(2 - \cos 2x)}$,then $27 I^2$ equals . . . . . . . .

Let $I_1 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}\sin (x)dx} $,$I_2 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}dx} $,and $I_3 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}(1 + x)\,dx} $. Consider the following statements:
$I: I_1 < I_2$
$II: I_2 < I_3$
$III: I_1 = I_3$
Which of the following is (are) true?

If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos^2 x}{1+e^x} dx = \pi(\alpha \pi^2 + \beta)$,where $\alpha, \beta \in \mathbb{Z}$,then $(\alpha + \beta)^2$ equals: