$\int_{-8}^{8} \frac{x^{5}+x^{3}}{4-x^{2}} \, dx = $

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\cos^{\frac{3}{2}} x}{\cos^{\frac{3}{2}} x + \sin^{\frac{3}{2}} x} \, dx = $ . . . . . . .

If $\int_0^{\pi / 2} \tan ^{14}\left(\frac{x}{2}\right) d x=2\left[\sum_{n=1}^7 f(n)-\frac{\pi}{4}\right]$,then $f(n)=$

For any integer $n,$ the integral $\int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x\,dx} = $

For $U_n = \int\limits_0^1 x^n (2 - x)^n \, dx$ and $V_n = \int\limits_0^1 x^n (1 - x)^n \, dx$,where $n \in N$,which of the following statements is true?

For $x \in \mathbb{R}$,let $f(x) = |\sin x|$ and $g(x) = \int_0^x f(t) \, dt$. Let $p(x) = g(x) - \frac{2}{\pi} x$. Then: