Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
不唯一
, 原函数集合
称不定积分, 记为 Initializing MathJax...
积分变量
, Initializing MathJax...
被积函数
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Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
被积函数中出现根号,优先考虑第二类换元法 出现平方差开跟、平方和开跟,考虑令x=sint或x=tant/sect,注意单调区间
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
适用范围:不同类型函数相乘
v'——易积分;u——易求导(反对幂指三)
有理函数积分方法:
Initializing MathJax...
Initializing MathJax...
若该极限存在且极限值与
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
几何意义:
Initializing MathJax...
曲边梯形面积的
代数和
微元法:①分割;②近似;③求和;④取极限;
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
称为Initializing MathJax...
在Initializing MathJax...
上的平均值Initializing MathJax...
习题:
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
换元要换限,换限要对应
Initializing MathJax...
例题:
华里士公式
:(点火公式)
Initializing MathJax...
当n为奇数时,=
Initializing MathJax...
当n为偶数时,=
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
连续 →Initializing MathJax...
可导.Initializing MathJax...
Initializing MathJax...
连续 →Initializing MathJax...
为Initializing MathJax...
的一个原函数Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
被积函数含变限积分时,则将其当作分部积分中的u
对谁求导,则变限积分被积函数中不能有谁出现
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
积分再现
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Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
设
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
对任意的
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
当
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
当
Initializing MathJax...
Initializing MathJax...
类似可定义:
Initializing MathJax...
若对任意的
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
瑕积分
(又称无界函数积分
), 称 Initializing MathJax...
瑕点
。当
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
当
Initializing MathJax...
Initializing MathJax...
类似可定义:当
Initializing MathJax...
Initializing MathJax...
一般分母为0的点为瑕点
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
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Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
以无穷区间上的反常积分为例
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
Initializing MathJax...
设
Initializing MathJax...
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Initializing MathJax...
Initializing MathJax...
注:
Initializing MathJax...
Initializing MathJax...
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Initializing MathJax...