$\int_0^\pi \frac{x \tan x}{\sec x + \tan x} \,dx = $

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?

The value of $\int_{-\pi/6}^{\pi/6} \left( \frac{\pi + 4x^{11}}{1 - \sin(|x| + \pi/6)} \right) dx$ is equal to: (in $\pi$)

For each positive integer $n$,define $f_n(x) = \min\left(\frac{x^n}{n!}, \frac{(1-x)^n}{n!}\right)$ for $0 \leq x \leq 1$. Let $I_n = \int_{0}^{1} f_n(x) dx$ for $n \geq 1$. Then,$\sum_{n=1}^{\infty} I_n$ is equal to

Let $g_i: \left[\frac{\pi}{8}, \frac{3\pi}{8}\right] \rightarrow \mathbb{R}, i=1, 2$,and $f: \left[\frac{\pi}{8}, \frac{3\pi}{8}\right] \rightarrow \mathbb{R}$ be functions such that $g_1(x)=1, g_2(x)=|4x-\pi|$ and $f(x)=\sin^2 x$,for all $x \in \left[\frac{\pi}{8}, \frac{3\pi}{8}\right]$.
Define $S_i = \int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} f(x) \cdot g_i(x) dx, i=1, 2$.
$(1)$ The value of $\frac{16S_1}{\pi}$ is.
$(2)$ The value of $\frac{48S_2}{\pi^2}$ is.

Let $p(x)$ be a function defined on $R$ such that $p'(x) = p'(1 - x)$ for all $x \in [0, 1]$,$p(0) = 1$,and $p(1) = 41$. Then $\int_{0}^{1} p(x) dx = $