定积分与反常积分
一、定积分概念
\[\begin{align}
&定义:设函数f(x)在区间[a,b]上有定义且有界\\
&(1)分割:将[a,b]分成n个[x_{i-1},x_{i}]小区间\\
&(2)求和:[x_{i-1},x_{i}]上取一点\xi_{i},\sum_{i=1}^{n}{f(\xi_{i})\Delta x_i},\lambda=\max{\Delta x_{1},\Delta x_{2},...,\Delta x_{n}}\\
&(3)取极限:若\lim_{\lambda \rightarrow 0}{\sum_{i=1}^{n}f(\xi_{i})\Delta x}\exist,且极值不依赖区间[a,b]分发以及点\xi_{i}的取法,则称f(x)在区间[a,b]上可积,\\
&\int^{b}_{a}{f(x)dx}=\lim_{\lambda \rightarrow 0}{f(\xi)\Delta x_{i}}
&\\
&注解:\\
&(1)\lambda \rightarrow0 \rightarrow \nleftarrow n\rightarrow \infty\\
&(2)定积分表示一个值,与积分区间[a,b]有关,与积分变化量x无关\\
&\int_{a}^{b}{f(x)dx}=\int_{a}^{b}{f(t)dt}\\
&(3)如果积分\int_{0}^{1}{f(x)dx}\exist,将[0,1]n等分,此时\Delta{x_{i}}=\frac{1}{n},取\xi_{i}=\frac{i}{n},\\
&\int_{0}^{1}f(x)dx=\lim_{\lambda \rightarrow 0}{\sum_{i=1}{n}{f(\xi_{i})\Delta x_{i}}}=\lim_{n\rightarrow \infty}\sum_{i=1}^{n}f(\frac{i}{n})\\
\end{align}
\]