$\int_{-1}^{1} x^{17} \cos^{4} x \, dx = $

If $a > 0$, then $\int_{-\pi}^{\pi} \frac{\sin^2 x}{1+a^x} dx$ is equal to

$\int\limits_{ - a}^a {f(x)\,dx} = $

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

If $F(x) = f(x) + f\left(\frac{1}{x}\right)$,where $f(x) = \int_{1}^{x} \frac{\log_{e} t}{1+t} dt$,then $F(e) = $

$\frac{3}{25} \int_0^{25 \pi} \sqrt{|\cos x - \cos^3 x|} \, dx =$