# 积分本文重定向自 積分

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x}$

${\displaystyle f(x)}$不定积分（或原函数）是指任何满足导数是函数${\displaystyle f(x)}$函数${\displaystyle F(x)}$。一个函数${\displaystyle f(x)}$的不定积分不是唯一的：只要${\displaystyle F(x)}$${\displaystyle f(x)}$的不定积分，那么与之相差一个常数的函数 ${\displaystyle F(x)+C}$也是${\displaystyle f}$的不定积分。本条目中主要介绍定积分，不定积分的介绍参见不定积分条目，无说明的情况下，下文中的“积分”一词均指“定积分”。

## 简介

${\displaystyle S=\int _{0}^{1}{\sqrt {x}}\,\mathrm {d} x\,\!.}$

${\displaystyle {\sqrt {0.2}}\left(0.2-0\right)+{\sqrt {0.4}}\left(0.4-0.2\right)+{\sqrt {0.6}}\left(0.6-0.4\right)+{\sqrt {0.8}}\left(0.8-0.6\right)+{\sqrt {1}}\left(1-0.8\right)\approx 0.7497.\,\!}$

${\displaystyle {\sqrt {\frac {0}{12}}}\left({\frac {1}{12}}-0\right)+{\sqrt {\frac {1}{12}}}\left({\frac {2}{12}}-{\frac {1}{12}}\right)+\cdots +{\sqrt {\frac {11}{12}}}\left(1-{\frac {11}{12}}\right)\approx 0.6203.\,\!}$

### 术语和标记

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x.}$

${\displaystyle \iint _{D}f(x,y)\,\!\,\mathrm {d} \sigma }$ 或者 ${\displaystyle \iint _{D}f(x,y)\,\!\,\mathrm {d} x\mathrm {d} y}$

## 严格定义

### 黎曼积分

${\displaystyle \sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})}$

${\displaystyle \left|\sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})-S\right|<\epsilon .\,}$

${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x.}$

### 勒贝格积分

${\displaystyle \int 1_{A}\,d\mu =\mu (A)}$

${\displaystyle \int f\,d\mu =\int \left(\sum _{i=1}^{n}a_{i}1_{A_{i}}\right)\,d\mu =\sum _{i=1}^{n}a_{i}\int 1_{A_{i}}\,d\mu =\sum _{i=1}^{n}a_{i}\mu (A_{i})}$:28

${\displaystyle \int f\,d\mu =\sup {\bigg \{}g,\quad g}$为简单函数，并且${\displaystyle f-g}$恒大于零${\displaystyle .\,{\bigg \}}}$:30

${\displaystyle \int f\,d\mu =\lim _{n\to +\infty }\left[\sum _{k=0}^{n2^{n}-1}{\frac {k}{2^{n}}}\mu \left({\frac {k}{2^{n}}}\leqslant f<{\frac {k+1}{2^{n}}}\right)+n\mu (f\geqslant n)\right]=\lim _{n\to +\infty }\left[{\frac {1}{2^{n}}}\sum _{k=0}^{n2^{n}-1}\mu \left({\frac {k}{2^{n}}}\leqslant f\right)\right]}$:344

${\displaystyle f^{+}:}$ 如果${\displaystyle f(x)\geqslant 0,}$${\displaystyle f^{+}(x)=f(x),}$ 否则${\displaystyle f^{+}(x)=0.}$
${\displaystyle f^{-}:}$ 如果${\displaystyle f(x)\leqslant 0,}$${\displaystyle f^{-}(x)=-f(x),}$ 否则${\displaystyle f^{-}(x)=0.}$

${\displaystyle \int _{A}f\,d\mu =\int f1_{A}\,d\mu .}$:345

### 其他定义

• 达布积分：等价于黎曼积分的一种定义，比黎曼积分更加简单，可用来帮助定义黎曼积分。
• 黎曼－斯蒂尔杰斯积分：黎曼积分的推广，用一般的函数g(x)代替x作为积分变量，也就是将黎曼和中的${\displaystyle (x_{i+1}-x_{i})}$推广为${\displaystyle (g(x_{i+1})-g(x_{i}))}$
• 勒贝格－斯蒂尔杰斯积分：勒贝格积分的推广，推广方式类似于黎曼－斯蒂尔杰斯积分，用有界变差函数g代替测度${\displaystyle \mu }$
• 哈尔积分：由阿尔弗雷德·哈尔于1933年引入，用来处理局部紧拓扑群上的可测函数的积分，参见哈尔测度
• 伊藤积分：由伊藤清于二十世纪五十年代引入，用于计算包含随机过程维纳过程半鞅的函数的积分。

## 性质

### 线性

${\displaystyle \int _{\mathcal {I}}(\alpha f+\beta g)=\alpha \int _{\mathcal {I}}f+\beta \int _{\mathcal {I}}g\,}$

${\displaystyle \int _{a}^{b}(\alpha f+\beta g)(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx+\beta \int _{a}^{b}g(x)\,dx.\,}$

${\displaystyle \int _{\mathcal {I}}(\alpha f+\beta g)\,d\mu =\alpha \int _{\mathcal {I}}f\,d\mu +\beta \int _{\mathcal {I}}g\,d\mu .}$

${\displaystyle \int _{a}^{c}f(x)\,dx=\int _{a}^{b}f(x)\,dx+\int _{b}^{c}f(x)\,dx\,}$

${\displaystyle \int _{{\mathcal {I}}\cup {\mathcal {J}}}f\,d\mu =\int _{\mathcal {I}}f\,d\mu +\int _{\mathcal {J}}f\,d\mu .}$

### 介值性质

${\displaystyle mL({\mathcal {I}})\leqslant \int _{\mathcal {I}}f\leqslant ML({\mathcal {I}})}$

### 绝对连续性

${\displaystyle \lim _{n\to \infty }\int _{{\mathcal {I}}_{n}}f(x)\,dx=\int _{\mathcal {I}}f(x)\,dx}$

### 积分不等式

${\displaystyle \left(\int _{\mathcal {I}}(fg)(x)\,dx\right)^{2}\leq \left(\int _{\mathcal {I}}f(x)^{2}\,dx\right)\left(\int _{\mathcal {I}}g(x)^{2}\,dx\right).}$

${\displaystyle \left|\int f(x)g(x)\,dx\right|\leqslant \left(\int \left|f(x)\right|^{p}\,dx\right)^{\frac {1}{p}}\left(\int \left|g(x)\right|^{q}\,dx\right)^{\frac {1}{q}}.}$

${\displaystyle \left(\int \left|f(x)+g(x)\right|^{p}\,dx\right)^{\frac {1}{p}}\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{\frac {1}{p}}+\left(\int \left|g(x)\right|^{p}\,dx\right)^{\frac {1}{p}}.}$

## 微积分基本定理

${\displaystyle F'(x)=f(x)}$

${\displaystyle F'(x)=f(x)}$

${\displaystyle f}$在区间[a, b]上的定积分满足：

${\displaystyle \int _{a}^{b}f(t)\mathrm {d} t=F(b)-F(a).}$

## 推广

### 反常积分

${\displaystyle \int _{0}^{\infty }{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=\pi }$

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=\lim _{\epsilon \to 0}\int _{a}^{b-\epsilon }f(x)\,\mathrm {d} x}$

${\displaystyle \int _{a}^{\infty }f(x)\,\mathrm {d} x=\lim _{b\to \infty }\int _{a}^{b}f(x)\,\mathrm {d} x}$

${\displaystyle I_{t}=\int _{1}^{t}{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=2\arctan {\sqrt {t}}-{\frac {\pi }{2}}}$

${\displaystyle \int _{1}^{\infty }{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=\lim _{t\to \infty }\int _{1}^{t}{\frac {dx}{(x+1){\sqrt {x}}}}={\frac {\pi }{2}}}$

${\displaystyle I_{s}=\int _{s}^{1}{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}={\frac {\pi }{2}}-2\arctan {\sqrt {s}}}$

${\displaystyle \int _{0}^{1}{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=\lim _{s\to 0}\int _{s}^{1}{\frac {dx}{(x+1){\sqrt {x}}}}={\frac {\pi }{2}}}$

${\displaystyle \int _{0}^{\infty }{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=\int _{0}^{1}{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}+\int _{1}^{\infty }{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}={\frac {\pi }{2}}+{\frac {\pi }{2}}=\pi }$

### 多重积分

${\displaystyle \int _{C}e^{-x^{2}-y^{2}}\,\mathrm {d} \sigma .}$

${\displaystyle \int _{C}e^{-x^{2}-y^{2}}\,\mathrm {d} \sigma =\int _{-1}^{1}\int _{-{\sqrt {1-y^{2}}}}^{\sqrt {1-y^{2}}}e^{-x^{2}-y^{2}}\,\mathrm {d} x\mathrm {d} y=\int _{0}^{2\pi }\int _{0}^{1}e^{-r^{2}}\,r\mathrm {d} r\mathrm {d} \theta .}$

## 参考来源

1. Robert G. Bartle. The Elements of Integration and Lebesgue Measure. Wiley Classics Library Edition. 1995. ISBN 978-0-471-04222-8 （英语）.
2. John K. Hunter, Bruno Nachtergaele. Applied Analysis. World Scientific（插图版）. 2001. ISBN 9789810241919 （英语）.
3. ^ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 1984, p. 56.