📚 Edexcel A-Level Pure Maths: Integration Techniques and Applications | 爱德思A-Level纯数学：积分技巧与应用
Integration is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It appears in Pure Year 1 and is extended significantly in Year 2 with advanced techniques such as substitution, integration by parts, partial fractions and applications to areas, volumes and differential equations. This article collects the key methods, common traps and efficient exam strategies.
积分是爱德思A-Level纯数学中考查最频繁的主题之一。它在纯数学第一年出现，在第二年大幅扩展，包括换元法、分部积分、部分分式以及面积、体积和微分方程等应用。本文汇总关键方法、常见陷阱和高效应考策略。
1. Standard Integrals and the Reverse Chain Rule | 标准积分与逆链式法则
Memorising the basic integrals is the foundation for every Edexcel integration question. The results must be automatic, including the constant of integration when no limits are given.
熟记基本积分公式是解答爱德思积分题的基础。这些结果必须能快速调用，不定积分不要漏写积分常数。
| f(x) | ∫ f(x) dx | 中文说明 |
|---|---|---|
| xⁿ, n ≠ −1 | xⁿ⁺¹/(n+1) + C | 幂函数积分 |
| eˣ | eˣ + C | 指数函数积分 |
| 1/x | ln|x| + C | 倒数函数积分 |
| sin x | −cos x + C | 正弦积分 |
| cos x | sin x + C | 余弦积分 |
| sec² x | tan x + C | 正割平方积分 |
The reverse chain rule is essentially integration by recognising a derivative product. If you see a composite function multiplied by the derivative of its inner function, you can write down the answer directly.
逆链式法则本质上是通过识别导数乘积来积分。如果你看到复合函数乘以其内层函数的导数，就可以直接写出结果。
∫ f'(g(x)) g'(x) dx = f(g(x)) + C
Example: ∫ 6x² cos(x³) dx = 2 sin(x³) + C, because the derivative of x³ is 3x² and 6x² = 2 × 3x².
例子：∫ 6x²
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