$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{2-\sin \theta}{2+\sin \theta}\right) d \theta$ is equal to

$\int_{-1}^1 \frac{\sqrt{1+x+x^2}-\sqrt{1-x+x^2}}{\sqrt{1+x+x^2}+\sqrt{1-x+x^2}} dx$ is equal to

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (x^{13} + x \cos x + \tan^{15} x + 1) \, dx$ is . . . . . .

Let $f(x)=\int_0^x g(t) \log _e\left(\frac{1-t}{1+t}\right) d t$,where $g$ is a continuous odd function. If $\int_{-\pi / 2}^{\pi / 2}\left(f(x)+\frac{x^2 \cos x}{1+e^x}\right) d x=\left(\frac{\pi}{\alpha}\right)^2-\alpha$,then $\alpha$ is equal to..............

If $f(x)$ is a continuous periodic function with period $T,$ then the integral $I = \int_a^{a + T} {f(x)\,dx} $ is

For any real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10, 10]$ by
$f(x) = \begin{cases} x - [x] & \text{if } [x] \text{ is odd} \\ 1 + [x] - x & \text{if } [x] \text{ is even} \end{cases}$
Then the value of $\frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos(\pi x) \, dx$ is