> 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/fan-chang-ji-fen-de-xi-tong-zhi-shi-ku.md).

# 反常积分的系统知识库

## 反常积分

### 无穷区间上的反常积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170208.png)

### 无界函数的反常积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170216.png)

## 常考题型与经典例题

### 反常积分的散、敛性

### 例1：判断反常积分散、敛

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170222.png)

### 例2：反常积分收敛的条件

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170229.png)

### 例3：判断反常积分散、敛

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170238.png)

### 例4：反常积分收敛的条件

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170244.png)

### 反常积分的计算

### 例5：第二换元法求反常积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170255.png)

### 例6：凑微分法求反常积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170301.png)

### 例7：分部积分法求反常积分

![](https://NothingLin.coding.net/p/picture/d/picture/git/raw/master/2022/1/22/20220122170306.png)