📚 Integration | A-Level CIE Maths Key Points Review | 积分考点精讲
Integration is a cornerstone of A-Level CIE Mathematics, bridging differentiation with real-world applications such as calculating areas, volumes, and solving differential equations. Mastering integration techniques—from basic antiderivatives to substitution and parts—is essential for success in both Pure Mathematics 1 and 3. This revision guide distils the key concepts, standard integrals, and common exam pitfalls to help you build confidence and accuracy.
积分是A-Level CIE数学的核心内容，它连接了微分与计算面积、体积、解微分方程等实际应用。无论是不定积分、定积分，还是换元法与分部积分，熟练掌握这些技巧对Pure Mathematics 1和3的考试至关重要。本文精讲考点、梳理标准积分公式，并提示常见易错点，助你扎实备考。
1. Introduction to Integration | 积分简介
Integration is the reverse process of differentiation. If differentiating F(x) gives f(x), then integrating f(x) recovers F(x) plus an arbitrary constant. It is used to find antiderivatives, compute net areas under curves, and solve differential equations that model exponential growth, decay, and motion.
积分是微分的逆运算。若F(x)的导数为f(x)，则对f(x)积分可还原出F(x)加一个任意常数。积分可用于求原函数、计算曲线下的净面积，以及求解描述指数增长、衰减和运动等的微分方程。
In CIE exams, integration questions range from simple ‘find the integral of …’ to multi-step problems involving substitution, parts, area, and volume of revolution. A strong grasp of fundamentals ensures you can handle any variation.
在CIE考试中，积分题从简单的“求……的积分”到涉及换元、分部积分、面积和旋转体体积的综合题不等。基础扎实才能应对灵活多变的题型。
2. Indefinite Integrals and Basic Rules | 不定积分与基本法则
The indefinite integral of a function f(x) is denoted by ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration. This constant must always be included for indefinite integrals.
函数f(x)的不定积分记作∫ f(x) dx = F(x) + C，其中F'(x)=f(x)，C为积分常数。做不定积分时切记加上常数C。
Basic algebraic rules simplify integration: ∫ k·f(x) dx = k ∫ f(x) dx for any constant k, and ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx. These allow term‑by‑term integration of polynomials and composite expressions.
基本运算法则包括：常数可提出积分号∫ k·f(x) dx = k∫ f(x) dx；函数和差的积分等于各自积分的和差∫ [f(x)±g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。这些法则使多项式及复合函数可以逐项积分。
3. Integration of Standard Functions | 标准函数的积分
Fluency with standard integrals saves time and reduces errors. The table below lists the must‑know integrals for CIE — commit them to memory.
熟练记忆标准积分能大幅提高解题速度与准确率。下表列出CIE必须掌握的常用积分，务必熟记。
| f(x) | ∫ f(x) dx |
| xⁿ (n ≠ -1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| aˣ (a>0) | aˣ/ln a + C |
| sin x | -cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
| cosec x cot x | -cosec x + C |
| sec x tan x | sec x + C |
| cosec² x | -cot x + C |
| 1/√(a²-x²) | sin⁻¹(x/a) + C |
| 1/(a²+x²) | (1/a) tan⁻¹(x/a) + C |
Always check by differentiating; this is your best safeguard against misremembered formulas.
每次积分后可通过求导
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