fundamental theorem of calculus

微积分基本定理

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For continuous real f on [a,b], the function G(x)=integral_a^x f(t) dt has derivative f(x) in (a,b); if F is any antiderivative of f on that interval, continuous at the endpoints, then integral_a^b f=F(b)-F(a).

实函数 f 在闭区间连续时，其变上限积分在内部的导数等于 f；若 F 是 f 的原函数且在端点连续，则定积分等于端点处 F 的差。

Example

For $f(t)=2t+1$, $G(x)=x^2+x$ on $[0,2]$, so $\int_0^2f=6$.

Usage and distinctions

The continuous-integrand version gives differentiation at every interior point; a general Lebesgue integrand yields an almost-everywhere statement.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Analysis in One Variable§4.2, Theorems 4.2.6–4.2.7 · Checked 2026-10-14

Translation sources

1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第一部分 一元微积分学，七、定积分：定积分概念与微积分基本定理 · Checked 2026-10-14

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