$$\int_0^1 \int_0^1 \left(p-(1-p)\sqrt{1-w}\exp(\frac{wx_{i}^2}{2})(1-\frac{1}{w+1}\exp(\frac{-w^2x_{i}^2}{2(w+1)}))\right) \cdot \prod_{j\neq i}^m(p+(1-p)\sqrt{1-w}\exp(\frac{wx_{j}^2}{2}))dw\,dp ,$$ where $0 \le w \le 1$, $0 \le p \le 1$ and $x_i \in \Bbb R \ \forall i = 1, \dots, m$.
I have been trying to show that this integration is monotonic in $x_{i}^2$ but I could not show it analytically. Numerically, I have shown that this integration is decreasing in $x_{i}^2$ by generating 100 $x_{i}$'s. Could anyone help me to show that the given integration is decreasing in $x_{i}^2$? Thanks in advance.
This is quite easy: we know that $t \mapsto \exp (t)$ increases, that $t \mapsto \exp (-t)$ decreases and that $-f$ increases / decreases whenever $f$ decreases / increases.
Taking $x_i ^2$ for $t$, we obtain that $(1-p) \sqrt {1-w} \exp \frac {w x_i ^2} 2$ and $- \frac 1 {w+1} \exp \frac {- w^2 x_i ^2} {2 (w + 1)}$ increase. Adding $1$ to the latter produces the increasing function $1 - \frac 1 {w+1} \exp \frac {- w^2 x_i ^2} {2 (w + 1)}$. The product of two increasing functions is increasing, so adding a minus sign in front of it produces the decreasing $- (1-p) \sqrt {1-w} (\exp \frac {w x_i ^2} 2) \left( 1 - \frac 1 {w+1} \exp \frac {- w^2 x_i ^2} {2 (w + 1)} \right)$. Adding a $p$ to it leaves it decreasing, and multiplying the result by the factor $\prod \limits _{j \ne i} \dots$ which doesn't depend on $x_i ^2$ doesn't alter its monotonicity, so the whole integrand is decreasing with respect to $x_i ^2$.
Finally, use the fact that if $(u,y) \mapsto f(u,y)$ is increasing /decreasing with respect to $u$, then $u \mapsto \int f(u, y) \ \Bbb d y$ will still be increasing / decreasing with respect to $u$.
(Alternatively, you could derive the integral with respect to $t = x_i ^2$ and show that the derivative is negative, but it would take longer.)