# 積分

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

${\displaystyle f}$不定積分（或稱原函數）是任何滿足其導函數是函數${\displaystyle f}$函數${\displaystyle F}$。一個函數${\displaystyle f}$的不定積分不是唯一的：只要${\displaystyle F}$${\displaystyle f}$的不定積分，那麼與之相差一個常數的函數 ${\displaystyle F+C}$也是${\displaystyle f}$的不定積分。本條目中主要介紹定積分，不定積分的介紹參見不定積分條目。

## 簡介

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

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

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

### 術語和標記

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

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

## 嚴格定義

### 黎曼積分

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

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

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

### 勒貝格積分

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

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

${\displaystyle \int f\,d\mu =\sup {\bigg \{}g,\quad g}$為簡單函數，並且${\displaystyle f-g}$恆大於零${\displaystyle .\,{\bigg \}}}$[1]:30

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

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

${\displaystyle \int _{A}f\,d\mu =\int f1_{A}\,d\mu .}$[2]:345

### 其他定義

• 達布積分：等價於黎曼積分的一種定義，比黎曼積分更加簡單，可用來幫助定義黎曼積分。
• 黎曼－斯蒂爾傑斯積分：黎曼積分的推廣，用一般的函數g(x)代替x作為積分變量，也就是將黎曼和中的${\displaystyle (x_{i+1}-x_{i})}$推廣為${\displaystyle (g(x_{i+1})-g(x_{i}))}$
• 勒貝格－斯蒂爾傑斯積分：勒貝格積分的推廣，推廣方式類似於黎曼－斯蒂爾傑斯積分，用有界變差函數g代替測度${\displaystyle \mu }$
• 哈爾積分：由阿爾弗雷德·哈爾於1933年引入，用來處理局部緊拓撲群上的可測函數的積分，參見哈爾測度
• 伊藤積分：由伊藤清於二十世紀五十年代引入，用於計算包含隨機過程維納過程半鞅的函數的積分。

## 性質

### 線性

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

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

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

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

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

### 介值性質

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

### 絕對連續性

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

### 積分不等式

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

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

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

## 微積分基本定理

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

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

${\displaystyle f}$在區間[a, b]上的定積分滿足：

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

## 推廣

### 反常積分

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

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

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

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

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

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

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

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

### 多重積分

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

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

## 參考來源

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