📚 A-Level Edexcel Maths: Integration by Substitution | A-Level Edexcel 数学：换元积分法
Integration by substitution is the reverse of the chain rule and is one of the most powerful techniques in the Edexcel A-Level Mathematics specification. This article walks through the core method, common choices of u, handling limits in definite integrals, and exam-style traps to avoid.
换元积分法是链式法则的逆向操作，也是 Edexcel A-Level 数学大纲中最有力的积分技巧之一。本文梳理核心方法、常见的 u 选择策略、定积分中积分限的处理方式以及考试中需要避免的常见陷阱。
1. The Core Idea: Reversing the Chain Rule | 核心思想：逆向链式法则
When you differentiate a composite function y = f(g(x)), the chain rule gives dy/dx = f'(g(x))·g'(x). Therefore, an integral of the form ∫ f'(g(x))·g'(x) dx can be worked out by setting u = g(x), so that du/dx = g'(x) and du = g'(x) dx. This transforms the integral into ∫ f'(u) du = f(u) + C = f(g(x)) + C.
当你对复合函数 y = f(g(x)) 求导时，链式法则给出 dy/dx = f'(g(x))·g'(x)。因此，形如 ∫ f'(g(x))·g'(x) dx 的积分可以通过设 u = g(x) 来计算，此时 du/dx = g'(x) 且 du = g'(x) dx。这样积分就变为 ∫ f'(u) du = f(u) + C = f(g(x)) + C。
∫ f'(g(x))·g'(x) dx = ∫ f'(u) du = f(u) + C
The key is to spot the inner function g(x) whose derivative g' the integrand. Once you choose u = g(x), the integral should become simpler in terms of u.
关键是要识别出内层函数 g(x)，并且它的导数 g'(x) 在被积函数中作为一个因子出现。一旦选择 u = g(x)，积分就应该用 u 表示后变得更简单。
2. Choosing u Strategically | 有策略地选择 u
Most Edexcel questions give a strong hint about the substitution, either in the question itself or through the structure of the integrand. As a general rule, u is usually the expression inside brackets, under a square root, in the denominator, or inside a trigonometric or exponential function.
大多数 Edexcel 考题都会通过题目本身或被积函数的结构给出很强的换元提示。一般来说，u 通常是括号内的表达式、根号下的表达式、分母中的表达式，或者三角函数、指数函数内部的表达式。
- u = (x² + 3) when integrating x·(x² + 3)⁵ dx
- u = sin x when integrating cos x·sin²x dx
- u = eˣ + 1 when integrating eˣ/(eˣ + 1) dx
- u = ln x when integrating (ln x)/x dx
- 被积函数为 x·(x² + 3)⁵ dx 时，设 u = x² + 3
- 被积函数为 cos x·sin²x dx 时，设 u = sin x
- 被积函数为 eˣ/(eˣ + 1) dx 时，设 u = eˣ + 1
- 被积函数为 (ln x)/x dx 时，设 u = ln x
The derivative du/dx should appear in the integrand, possibly missing only a constant multiple. You can always adjust for a constant factor because constants can move outside the integral.
导数 du/dx 应该出现在被积函数中，可能只差一个常数倍数。你总是可以调整常数因子，因为常数可以移到积分号外。
3. Standard Pattern: f'(x)·[f(x)]ⁿ | 标准模式：f'(x)·[f(x)]ⁿ
One of the most common Edexcel patterns is ∫ f'(x)·[f(x)]ⁿ dx. If you set u = f(x), then du = f'(x) dx, and the integral becomes ∫ uⁿ du = uⁿ⁺¹/(n+1) + C.
Edexcel 最常见的模式之一是 ∫ f'(x)·[f(x)]ⁿ dx。如果设 u = f(x)，则 du = f'(x) dx，积分就变为 ∫ uⁿ du = uⁿ⁺¹/(n+1) + C。
Example: Evaluate ∫ 2x·(x² + 5)⁴ dx.
例子：计算 ∫ 2x·(x² + 5)⁴ dx。
, so du/dx = 2x and du = 2x dx. The integral becomes ∫ u⁴ du = u⁵/5 + C = (x² + 5)⁵/5 + C.
设 u = x² + 5，则 du/dx = 2x，所以 du = 2x dx。积分变为 ∫ u⁴ du = u⁵/5 + C = (x² + 5)⁵/5 + C。
of (x² + 5)⁵/5 is (1/5)·5(x² + 5)⁴·2x = 2x(x² + 5)⁴, which matches the original integrand.
一定要通过求导来检查答案。(x² + 5)⁵/5 的导数是 (1/5)·5(x² + 5)⁴·2x = 2x(x² + 5)⁴，与原始被积函数一致。
4. Using u = ax + b for Linear Inner Functions | 内层函数为线性时使用 u = ax + b
ion is straightforward. The derivative du/dx is a constant, so you only need to divide by that constant.
当内层函数是线性时，例如 u = 3x + 2 或 u = 5 − 4x，换元非常简单。导数 du/dx 是一个常数，因此你只需要除以这个常数。
Example: Evaluate ∫ (2x + 1)⁶ dx.
例子：计算 ∫ (2x + 1)⁶ dx。
2 and dx = du/2. The integral becomes ∫ u⁶·(1/2) du = (1/2)·(u⁷/7) + C = (2x + 1)⁷/14 + C.
设 u = 2x + 1，则 du/dx = 2，所以 dx = du/2。积分变为 ∫ u⁶·(1/2) du = (1/2)·(u⁷/7) + C = (2x + 1)⁷/14 + C。
This technique also applies to integrands such as sin(ax + b), cos(ax + b), and eᵃˣ⁺ᵇ. In each case, the antiderivative includes division by the coefficient a.
这个技巧也适用于 sin(ax + b)、cos(ax + b) 和 eᵃˣ⁺ᵇ 等被积函数。在每种情况下，原函数都包含除以系数 a。
5. Definite Integrals and Changing Limits | 定积分与积分限变换
For definite integrals, you have two valid methods. Method 1 is to find the indefinite integral in terms of x first, then substitute the original limits. Method 2 is to change the limits when you substitute u, so you never return to x.
对于定积分，有两种有效方法。方法一是先求出用 x 表示的不定积分，然后代入原始积分限。方法二是在换元时同时改变积分限，这样就不必再回到 x。
Example: Evaluate ∫ from 0 to 1 of 2x·(x² + 1)³ dx using changed limits.
例子：使用变换积分限的方法计算 ∫ 从 0 到 1 的 2x·(x² + 1)³ dx。
= 2. The integral becomes ∫ from 1 to 2 of u³ du = [u⁴/4] from 1 to 2 = 16/4 − 1/4 = 15/4.
设 u = x² + 1，则 du = 2x dx。当 x = 0 时，u = 1；当 x = 1 时，u = 2。积分变为 ∫ 从 1 到 2 的 u³ du = [u⁴/4] 从 1 到 2 = 16/4 − 1/4 = 15/4。
Changing limits is usually faster, but many candidates forget to change them and incorrectly substitute x-values into a u-expression. Always write down the new limits clearly.
变换积分限通常更快，但许多考生忘记改变积分限，错误地将 x 值代入 u 的表达式中。一定要清楚地写出新的积分限。
6. Trigonometric Substitutions | 三角换元
Trigonometric integrals often follow the pattern ∫ f'(x)·sin(f(x)) dx or ∫ f' be solved by setting u = sin x, giving du = cos x dx and ∫ u² du = u³/3 + C = sin³x/3 + C.
三角积分通常遵循 ∫ f'(x)·sin(f(x)) dx 或 ∫ f'(x)·cos(f(x)) dx 的模式。例如，∫ cos x·sin²x dx 可以通过设 u = sin x 来求解，得到 du = cos x dx，积分变为 ∫ u² du = u³/3 + C = sin³x/3 + C。
with u = tan x, since du/dx = sec²x. The integral becomes ∫ u du = u²/2 + C = tan²x/2 + C.
类似地，∫ sec²x·tan x dx 可以通过设 u = tan x 来求解，因为 du/dx = sec²x。积分变为 ∫ u du = u²/2 + C = tan²x/2 + C。
u = cos x, then du = −sin x dx. The negative sign must be carried through the calculation.
注意符号：如果 u = cos x，那么 du = −sin x dx。负号必须贯穿整个计算过程。
7. Substitution with Exponential Functions | 指数函数换元
For integrands involving eˣ, a useful substitution is often u = eˣ + a or u = eˣ. In the integral ∫ eˣ/(eˣ + 2) dx, setting u = eˣ + 2 gives du = eˣ dx, so the integral becomes ∫ 1/u du = ln|u| + C = ln|eˣ + 2| + C.
对于包含 eˣ 的被积函数，通常有用的换元是 u = eˣ + a 或 u = eˣ。在积分 ∫ eˣ/(eˣ + 2) dx 中，设 u = eˣ + 2，则 du = eˣ dx，因此积分变为 ∫ 1/u du = ln|u| + C = ln|eˣ + 2| + C。
Another common type is ∫ x·ex² u = x², so du = 2x dx and x dx = du/2. The integral is (1/2)∫ eᵘ du = (1/2)eᵘ + C = (1/2)ex² + C.
另一种常见类型是 ∫ x·ex² dx。设 u = x²，则 du = 2x dx，所以 x dx = du/2。积分为 (1/2)∫ eᵘ du = (1/2)eᵘ + C = (1/2)ex² + C。
Because eᵘ is its own derivative, exponential substitutions often produce elegant antiderivatives that are easy to check.
因为 eᵘ 的导数就是它本身，指数换元通常会产生优雅的原函数，而且很容易检验。
8. Common Mistakes to Avoid | 需要避免的常见错误
One of the biggest mistakes is forgetting to replace dx with du in terms of u. If du/dx is not simply a constant, candidates sometimes write du instead of dx without dividing by the derivative.
最大的错误之一是忘记用 u 表示 du 来替换 dx。如果 du/dx 不只是一个常数，考生有时会直接写 du 而不除以导数。
Another frequent error is changing the variable but not changing the limits in a definite integral. The final answer should not mix x-limits with a u-antiderivative.
另一个常见错误是在定积分中更换了变量却没有更换积分限。最终答案不应把 x 积分限和 u 的原函数混在一起。
- Forgetting the constant multiple from du/dx
- Missing a negative sign when du/dx is negative
- Not returning to the original variable in an indefinite integral
- Choosing u so that the new integral is harder, not easier
- 忘记 du/dx 带来的常数倍数
- 当 du/dx 为负时漏掉负号
- 在不定积分中没有换回原变量
- 选择的 u 让新积分更难而不是更简单
ep: u = ..., du/dx = ..., dx = ... du, then rewrite the whole integral before integrating.
练习写出每一步：u = ...，du/dx = ...，dx = ... du，然后在积分前重写整个积分。
9. Exam Tips and Worked Practice | 考试技巧与练习
In Edexcel exams, substitution questions are often worth 5 to 7 marks. Method marks are awarded for the correct choice of u, finding du/dx, and replacing all parts of the integral.
在 Edexcel 考试中，换元题通常占 5 到 7 分。方法分会给在正确选择 u、求出 du/dx 以及替换积分中的所有部分。
Worked practice: Evaluate ∫ x·√x² + 4 dx.
练习：计算 ∫ x·√x² + 4 dx。
Let u = x² + 4, then du/dx = 2x, so x dx = du/2. The integral becomes (1/2)∫ √u du = (1/2)·(2/3)u3/2 + C = (1/3)(x² + 4)3/2 + C.
设 u = x² + 4，则 du/dx = 2x，所以 x dx = du/2。积分变为 (1/2)∫ √u du = (1/2)·(2/3)u3/2 + C = (1/3)(x² + 4)3/2 + C。
Worked practice: Evaluate ∫ from 0 to π/2 of cos x·sin³x dx.
练习：计算 ∫ 从 0 到 π/2 的 cos x·sin³x dx。
en x = π/2, u = 1. The integral becomes ∫ from 0 to 1 of u³ du = [u⁴/4] from 0 to 1 = 1/4.
设 u = sin x，则 du = cos x dx。当 x = 0 时，u = 0；当 x = π/2 时，u = 1。积分变为 ∫ 从 0 到 1 的 u³ du = [u⁴/4] 从 0 到 1 = 1/4。
Always stop and check whether the integral is indefinite or definite before deciding how to handle u and the limits.
在决定如何处理 u 和积分限之前，一定要停下来检查这是不定积分还是定积分。
10. Summary and Quick Revision Table | 小结与快速复习表
Use the table below as a rapid revision checklist for Edexcel substitution questions.
使用下表作为 Edexcel 换元题的快速复习清单。
| Integral Form | Substitution | Result Pattern |
|---|---|---|
| ∫ f'(x)·[f(x)]ⁿ dx | u = f(x) | [f(x)]ⁿ⁺¹/(n+1) + C |
| ∫ f'(x)·sin(f(x)) dx | u = f(x) | −cos(f(x)) + C |
| ∫ f'(x)·cos(f(x)) dx | u = f(x) | sin(f(x)) + C |
| ∫ f'(x)·e^(f(x)) dx | u = f(x) | e^(f(x)) + C |
| ∫ f'(x)/f(x) dx | u = f(x) | ln|f(x)| + C |
Remember that integration by substitution is about recognising the derivative of an inner function inside the integrand. The more patterns you practise, the faster you will spot the right substitution in the exam.
请记住，换元积分法就是识别被积函数中内层函数的导数。练习的模式越多，你在考试中就能越快发现正确的换元。
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