Integration Part 1: Reversing Differentiation and Indefinite Integrals | 积分第一部分：微分逆运算与不定积分

📚 Integration Part 1: Reversing Differentiation and Indefinite Integrals | 积分第一部分：微分逆运算与不定积分

In A-Level Mathematics, integration is often introduced as the reverse process of differentiation. Part 1 focuses on indefinite integrals, where we recover a family of functions whose derivative is the given expression. This article develops the notation, power rule, constant of integration, and the key skill of finding f(x) from f'(x) that exam papers frequently test.

在 A-Level 数学中，积分通常作为微分的逆运算引入。第一部分重点讲解不定积分，即由已知导数恢复一族原函数。本文将系统整理积分符号、幂函数积分法则、积分常数，以及考试中常考的由 f'(x) 求 f(x) 的核心技能。

1. What Is Integration? | 什么是积分？

Integration answers the question: given the derivative of a function, what was the original function? If dy/dx = 2x, then y could be x² because d/dx(x²) = 2x.

积分回答的问题是：已知某个函数的导数，原来的函数是什么？如果 dy/dx = 2x，那么 y 可以是 x²，因为 d/dx(x²) = 2x。

However, x² + 1, x² − 3 and x² + 100 also differentiate to 2x. Integration therefore gives a family of functions, not just one function.

然而，x² + 1、x² − 3、x² + 100 的导数也都是 2x。因此积分得到的是一个函数族，而不是唯一的一个函数。

In Part 1, we work with indefinite integrals, which do not have fixed limits

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