不定积分

一、不定积分的概念

原函数

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-1.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-2.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-3.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-4.png" width="500" />

不定积分定义

• 所有原函数就是不定积分

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-5.png" width="500" />

• 求不定积分就是求所有原函数

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-6.png" width="500" />

• 微分和积分互为逆运算, 对于一个函数f(x)先求积分再求积分等于本身f(x)，　先求微分再求积分等于f(x)+C (C是常数，注意：不定积分必须有C)

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-7.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-8.png" width="500" />

三，不定积分的计算

基本公式

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-9.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-10.png" width="500" />

不定积分的性质

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-11.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-12.png" width="500" />

• 直接积分法( 就是找规律，向不定积分性质靠拢)

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-13.png" width="500" />

第一类换元法

• 直接凑（凑微分法）

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-14.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-15.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-16.png" width="500" />

• 间接法（先凑微分，在凑公式）

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-17.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-18.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-19.png" width="500" />

第二换元法　（简单理解）

• 根式换元

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-20.png" width="500" />

• 三角换元 ( 了解就可以了 )

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-21.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-22.png" width="500" />

分部积分法

• 分部积分法则

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-23.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-24.png" width="500" />

• 分部积分u,v‘的确认　（注意：这是分部积分的重点也是难点）

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-25.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-26.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-27.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-28.png" width="500" />

简单有理函数求微分

• 简单有理函数概念

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-29.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-30.png" width="500" />

• 简单有理函数求微分的方法：先分项，后微分

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-jifen-31.png" width="500" />