微积分基本定理的准确陈述

在我们国内，微积分教科书往往微积分基本定理说成是“牛顿-”莱布尼兹公式”，而对该定理的精神实质避而不谈。

实质上，这个重要定理的核心思想是：定积分的数值计算可以用“符号积分法”技巧来代替，节省大量的数字计算成本。

该定理第一部分证明了处处连续的被积分函数的原函数存在性，指出上限可变的定积分所确定函数就是所要求一个原函数。，

原函数通过符号积分法的技巧进行形式计算获得，大大方便了定积分的数值计算成本。为此发现，牛顿-莱布尼兹的声誉永留人间。

请看原文：

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions.

Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by symbolic integration allows for avoiding numerical integration（这是要点！） to compute integrals.（全文完）

袁萌  3月12日