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§1 不定积分的概念与性质

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• 定义1设函数f 与F 在区间I上有定义，若

• 原函数举例

• 提问：

• （1）什么条件下，一个函数的原函数存在？

( 2 )如果f (x)有原函数，一共有多少个？

( 3 )任意两个原函数之间有什么关系？

• 几点说明：

1°原函数存在定理：连续函数一定有原函数.

2°若F'(x) = f (x)，则对任意常数C， F(x)+C都是

f (x)的原函数. 如(sin x)cos x , 则 (sin x+C)cos x .

3°设G(x) 、F(x)是 f (x)的任意两个原函数.

• 定义2f (x)在区间I上全体原函数成为 f 在 I上的不定积分. 记作

• 结论：求f (x)的不定积分只要求它的一个原函数F(x)再加任意常数C.

2x的积分曲线

• 不定积分的几何意义

F(x)+c的图形是由F(x)的图形沿 y 轴平移c(任意的）所得积分曲线组成的曲线轴.

• 性质1

• 性质2

( k为常数 k≠0）

• 性质3