# 不定积分的系统知识库

## 一、不定积分的概念与性质

### 原函数

![](https://i.loli.net/2021/11/09/X27jkTtDQLwvBE9.png)

### 不定积分的定义

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109010219.png)

### 不定积分几何意义

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109010345.png)

### 原函数存在定理

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109010522.png)

###

### 例题：原函数的存在判断

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109010700.png)

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109010757.png)

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109010938.png)

## 二、不定积分的性质

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109011030.png)

### 不定积分公式

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109011111.png)

### 例2:根据公式法则求不定积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109011334.png)

## 三、三种主要积分法

### 换元积分法

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109011541.png)

### 常见的凑微分形式

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109011635.png)

### 例3：根据凑微分法求不定积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109011909.png)

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109011949.png)

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109012039.png)

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/9/20211109012121.png)

### 例4：根据凑微分法求不定积分

![](https://i.loli.net/2021/11/10/DCH1LhbB32PkiAv.png)

### 例5：根据凑微分法求不定积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/10/20211110004348.png)

### 第二换元积分法

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/10/20211110004501.png)

### 例6：根据第二换元积分法求不定积分

#### T1：

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/10/20211110004640.png)

#### T2：

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/10/20211110004754.png)

#### T3

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/11/20211111012007.png)

#### T4

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/11/20211111012052.png)

### 分部积分法

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/11/20211111012203.png)

### 例8：分部积分法求不定式积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/11/20211111165930.png)

### 例7：分部积分法求不定式积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/11/20211111170101.png)

### 例9：分部积分法求不定积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/11/20211111172831.png)

## 四、三类常见可积函数积分

### 有理函数积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/12/20211112200826.png)

### 例10：有理函数积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/12/20211112201019.png)

### 例11：有理函数积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/12/20211112201123.png)

### 例12：有理函数积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/12/20211112201230.png)

### 三角有理式积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/13/20211113172008.png)

### 例13：三角有理式求积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/13/20211113172142.png)

### 例14：三角有理式求积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/13/20211113174404.png)

### 例15：三角有理式求积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/13/20211113174803.png)

### 例16：三角有理式求积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/13/20211113175121.png)

### 例17：简单无理式求积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/13/20211113175318.png)

## 经典例题与常考题型

### 例18：简单无理式求积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115005306.png)

### 例19：求分段函数积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115005420.png)

### 例20：不定积分中的常数问题

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115005618.png)

### 例21：第二换元积分法求无理式积分

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115005803.png)

### 例22：分部积分法求

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115005934.png)

### 例23：换元积分法与分部积分法

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115010517.png)

### 例24：换元积分法与分部积分法

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115010702.png)

### 例25：分部积分法

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115010818.png)

### 例26：分部积分法

![](https://nothinglin.coding.net/p/picture/d/picture/git/raw/master/2021/11/15/20211115010914.png)