Integration of Power Functions: The ‘Add One to the Power’ Rule | 幂函数积分：指数加一法则

📚 Integration of Power Functions: The ‘Add One to the Power’ Rule | 幂函数积分：指数加一法则

In this revision guide, we explore one of the most fundamental rules in calculus: integrating power functions of the form xⁿ. This rule is often summarised as ‘add one to the power and divide by the new power’, but it has a critical exception that is just as important. We will develop the rule from first principles, apply it to a range of worked examples, and highlight the pitfalls that commonly appear in A-Level examinations.

在本复习指南中，我们探讨微积分中最基本的法则之一：对形如 xⁿ 的幂函数进行积分。这条法则常被概括为“指数加一，然后除以新的指数”，但它有一个同样重要的例外情况。我们将从基本原理出发推导该法则，应用于多个例题，并指出 A-Level 考试中常见的陷阱。

1. The Basic Rule | 基本法则

For any real constant n ≠ -1, the indefinite integral of xⁿ is given by:

对于任意实常数 n ≠ -1，xⁿ 的不定积分公式为：

∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C

This result follows directly from reversing the derivative of a power function. Because the derivative of xⁿ⁺¹ is (n+1)xⁿ, dividing by (n+1) gives exactly xⁿ as the derivative.

该结果直接来自于导数运算的逆运算。因为 xⁿ⁺¹ 的导数是 (n+1)xⁿ，除以 (n+1) 后恰好得到 xⁿ 的导数。

2. Why ‘Add One to the Power’ Works | 为什么“指数加一”能成立

Start with the function y = xⁿ⁺¹. Its derivative is:

从函数 y = xⁿ⁺¹ 开始，其导数为：

dy/dx = (n + 1)xⁿ

If we divide both sides by (n+1), we obtain dy/dx = xⁿ for the function y = xⁿ⁺¹/(n+1). Hence the integral of xⁿ is exactly that divided expression, plus a constant.

如果两边同时除以 (n+1)，则对于函数 y = xⁿ⁺¹/(n+1)，其导数恰好为 xⁿ。因此 xⁿ 的积分正是这个除以后的表达式，再加上一个常数。

3. The Constant of Integration | 积分常数

Since the derivative of any constant is zero, two functions that differ only by a constant have the same derivative. When we integrate, we must therefore include ‘+ C’ to represent all possible antiderivatives.

由于任何常数的导数都是零，仅相差一个常数的两个函数具有相同的导数。因此积分时必须在原函数后面加上 “+ C” 来表示所有可能的原函数。

For example, ∫ 2x dx = x² + C. Both x² and x² + 5 have derivative 2x, so C absorbs any constant shift.

例如，∫ 2x dx = x² + C。x² 和 x² + 5 的导数都是 2x，所以 C 包含了所有常数平移。

4. Worked Example 1: Simple Powers | 例题1：简单幂函数

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