> For the complete documentation index, see [llms.txt](https://nothinglin.gitbook.io/kao-yan-bi-ji/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nothinglin.gitbook.io/kao-yan-bi-ji/wu-ding-ji-fen/ding-ji-fen-de-xi-tong-zhi-shi-ku.md).

# 定积分的系统知识库

## 定积分

### 定积分的概念

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020724.png)

### 定积分存在的充分条件

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020737.png)

### 定积分的几何意义

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020743.png)

### 定积分的性质

### 1.不等式性质

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020748.png)

### 2.中值定理

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020752.png)

### 积分上限函数

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020757.png)

### 1.积分上限函数（重要定理）

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020803.png)

### 定积分的计算

### 1.牛顿-莱布尼茨公式

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020808.png)

### 2.定积分换元积分法

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020812.png)

### 3.定积分分部积分法

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020817.png)

### 4.利用周期性和奇偶性求定积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020821.png)

### 5.定积分特性公式求定积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020826.png)

## 经典例题与常考题型

### 定积分的概念、性质及几何意义

### 例1：定积分的定义求极限

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020832.png)

### 例2：定积分的定义求极限

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020838.png)

### 例3：定积分的定义求极限

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020845.png)

### 例4：定积分的定义求极限

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020850.png)

### 例5：积分上限函数的概念应用

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020855.png)

### 例6：定积分的几何意义

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020906.png)

### 例7：定积分的几何意义

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020914.png)

### 例8：定积分的几何意义

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020919.png)

### 例9：定积分的中值定理

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020923.png)

### 定积分的计算

### 例10：利用函数奇偶性求定积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020929.png)

### 例11：利用函数奇偶性+几何方式求定积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020935.png)

### 例12：利用函数几何意义来求定积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020940.png)

### 例13：三角降幂+分部积分法

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020944.png)

### 例14：无理式求定积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020949.png)

### 例15：无理式求定积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/20/20211120020954.png)

### 例16：定积分+arcsinx题型+分部积分法

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004334.png)

### 例17：积分上限函数+定积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004344.png)

### 变上限积分

### 例18：求变上限积分函数

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004349.png)

### 例19：变上限积分函数的应用

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004357.png)

### 例20：求变上限积分函数

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004402.png)

### 例21：分段函数+变上限积分函数

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004409.png)

### 例22：变上限积分函数+绝对值+分段函数

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004416.png)

### 例23：变上限积分函数+可导+断点+连续

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004425.png)

### 例24：求极限+变上限积分函数

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004432.png)

### 例25：求极限+变上限积分函数+无理式

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004438.png)

### 例26：变上限积分函数的应用

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2021/11/25/20211125004445.png)