Integration: The Power of Each Term Increases by 1 | 积分：每项幂次增加1

📚 Integration: The Power of Each Term Increases by 1 | 积分：每项幂次增加1

Integration is one of the two fundamental operations in calculus and a cornerstone of the Edexcel A-Level Mathematics syllabus. Whether you are computing areas under curves, solving differential equations, or finding volumes of revolution, the power rule of integration is the tool you will use more than any other. At its heart lies a beautifully simple idea: when you integrate a term of the form xⁿ, the power of that term increases by 1.

积分是微积分中的两大基本运算之一，也是 Edexcel A-Level 数学大纲的基石。无论是计算曲线下的面积、求解微分方程，还是求旋转体的体积，积分幂法则都是你最常用的工具。其核心思想简洁而优雅：对形如 xⁿ 的项进行积分时，该项的幂次增加1。

This article will guide you through the essential techniques of integration, with a special focus on why and how the power increases by 1 in every term. We will explore worked examples, common pitfalls, and exam-style questions to ensure you are fully prepared for the Edexcel examinations.

本文将引导你掌握积分的核心技巧，特别聚焦于为什么以及如何使每一项的幂次增加1。我们将通过例题、常见陷阱和考试风格题目，确保你为 Edexcel 考试做好充分准备。

1. The Power Rule: n → n+1 | 幂法则：n → n+1

The power rule of integration states that for any real number n ≠ -1:

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C

The most important observation is that the exponent increases from n to n+1. This is why integration is sometimes described as the reverse of differentiation: when differentiating, the power decreases by 1; when integrating, the power increases by 1. For example, differentiating x³ gives 3x², while integrating 3x² gives x³ + C — the power has gone up from 2 back to 3.

积分幂法则指出：对于任何实数 n ≠ -1：

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C

最重要的观察是指数从 n 增加到 n+1。这就是为什么积分有时被称为微分的逆运算：微分时幂次减1，积分时幂次加1。例如，对 x³ 求导得 3x²，而对 3x² 积分得 x³ + C——幂次从2重新回到了3。

The constant C, known as the constant of integration, is essential. Since the derivative of any constant is zero, infinitely many functions differ only by a constant yet share the same derivative. The Edexcel mark scheme always awards a mark for including + C when integrating indefinitely, so do not forget it.

常数 C 称为积分常数，必不可少。由于任何常数的导数都为零，因此无数个仅相差一个常数的函数拥有相同的导数。Edexcel 评分标准中，不定积分写出 + C 才能获得对应分数，所以千万不要遗漏。

2. Why Does the Power Increase by 1? | 为什么幂次增加1？

The answer lies in the relationship between integration and differentiation. Recall that when we differentiate xⁿ⁺¹, the power n+1 is brought down as a coefficient:

d/dx (xⁿ⁺¹ / (n+1)) = (n+1) · xⁿ / (n+1) = xⁿ

By dividing by n+1, the coefficient cancels perfectly, leaving exactly xⁿ. This confirms that xⁿ⁺¹ / (n+1) is indeed an antiderivative of xⁿ. The entire construction is designed so that the derivative of the result gives back the original function.

答案在于积分与微分之间的关系。回顾：对 xⁿ⁺¹ 求导时，幂次 n+1 作为系数被提出：

d/dx (xⁿ⁺¹ / (n+1)) = (n+1) · xⁿ / (n+1) = xⁿPublished by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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