數學: 積分整理

前言

此篇博客用來整理一些技巧結論性質的積分

[1] $\int_{x_1}\int_{x_2}[f(x_1)+f(x_2)]p(x_1,x_2)dx_1dx_2=\int_{x_1}f(x_1)p(x_1)dx_1+\int_{x_2}f(x_2)p(x_2)dx_2$
證明：
$\begin{aligned} \int_{x_1}\int_{x_2}[f(x_1)+f(x_2)]p(x_1,x_2)dx_1dx_2&=\int_{x_1}[f(x_1)\int_{x_2}p(x_1,x_2)dx_2]dx_1 + \int_{x_2}[f(x_2) \int_{x_1}p(x_1,x_2)dx_1]dx_2 \\ &=\int_{x_1}f(x_1)p(x_1)dx_1+\int_{x_2}f(x_2)p(x_2)dx_2 \end{aligned}$