📚 Integration Techniques for Prelims Paper I-III: Chapter 9C Homework | 预科试卷 I-III：第9C章作业 – 积分技巧
Integration is a core topic in A-level Pure Mathematics, appearing across Papers I, II and III. In Chapter 9C homework, students often meet indefinite integrals, definite integrals, areas between curves, and the first formal methods such as substitution and integration by parts. This revision guide gives you the exact tools needed to complete the homework accurately and to prepare for Prelims-style questions.
积分是 A-level 纯数学的核心主题，出现在试卷 I、II 和 III 中。在第 9C 章作业中，学生通常会遇到不定积分、定积分、曲线之间的面积，以及换元积分法和分部积分法等首个正式方法。本复习指南为你提供完成作业并准备预科考试题型所需的精确工具。
1. Learning Objectives | 学习目标
The Chapter 9C homework on integration tests three main skills: reversing differentiation, evaluating definite integrals, and applying integration to find areas. It also introduces the first advanced techniques, substitution and integration by parts, which appear in Papers II and III.
第 9C 章积分作业测试三项主要技能：逆转微分、计算定积分，以及应用积分求面积。它还介绍了最初的进阶技巧——换元积分法和分部积分法，这些出现在试卷 II 和 III 中。
- Integrate powers, exponential and trigonometric functions. 对幂函数、指数函数和三角函数积分。
- Evaluate definite integrals using antiderivatives. 使用原函数计算定积分。
- Find areas under curves and between curves. 求曲线下方及曲线之间的面积。
- Apply substitution and integration by parts. 应用换元积分法和分部积分法。
2. The Basic Integration Rules | 基本积分法则
The power rule is the foundation of most Chapter 9C questions. You must remember the exception n = −1, where the integral is the natural logarithm of the absolute value of x.
幂法则是大多数第 9C 章题目的基础。你必须记住例外情况 n = −1，此时积分为 x 的绝对值的自然对数。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
| Function | Integral | 中文 |
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C | ln|x
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