Integration by Substitution: Edexcel A-Level Pure Maths Guide | 代换积分法：Edexcel A-Level 纯数指南

📚 Integration by Substitution: Edexcel A-Level Pure Maths Guide | 代换积分法：Edexcel A-Level 纯数指南

This article, compiled from a comprehensive Edexcel revision PDF (Joiner Section 4, page 226), provides an in-depth exploration of integration by substitution, a fundamental technique in A-Level Pure Mathematics. Mastery of this method is essential for tackling complex integrals and securing high marks in the exam.

本文取材自一份全面的Edexcel复习PDF汇编（合并文件第4部分，第226页），深入探讨代换积分法这一A-Level纯数中的基本技巧。掌握该方法对于攻克复杂积分并在考试中取得高分至关重要。

1. The Core Idea of Substitution | 代换法的核心思想

Integration by substitution is the reverse process of the chain rule for differentiation. It allows us to transform a complicated integral into a simpler one by changing the variable.

代换积分法是微分链式法则的逆过程。它通过改变变量，将复杂的积分转化为更简单的形式。

Suppose we have an integral of the form ∫ f(g(x))g'(x) dx. By letting u = g(x), then du/dx = g'(x) ⇒ du = g'(x) dx. The integral becomes ∫ f(u) du.

假设我们有形如 ∫ f(g(x))g'(x) dx 的积分。令 u = g(x)，则 du/dx = g'(x) ⇒ du = g'(x) dx。积分变为 ∫ f(u) du。

The key is to spot an 'inner function' and its derivative appearing as a factor. Once the substitution is made, the integral often reduces to a standard form.

关键在于识别一个'内层函数'及其导数作为一个因子出现。一旦完成代换，积分通常会化为标准形式。

∫ f(g(x))g'(x) dx = ∫ f(u) du, where u = g(x)

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2. Recognising When to Substitute | 识别何时使用代换法

the inner function is present, perhaps missing only a constant factor. For example, in ∫ x√x²+1 dx, the inner function is x²+1 and its derivative 2x is closely related to the x term.

寻找一个复合函数，其内层函数的导数出现在被积函数中，或许只相差一个常数因子。例如，在 ∫ x√x²+1 dx 中，内层函数是 x²+1，其导数 2x 与 x 项密切相关。

Another strong clue is when the integral involves functions like sin(ax+b), e^kx, or ln(x) mixed with algebraic terms that resemble their derivatives.

另一个明显的线索是当积分包含诸如 sin(ax+b)、e^kx 或 ln(x) 等函数，并且与类似其导数的代数项混合时。

If an integrand can be written as (something)ⁿ the derivative of that something, substitution with u = that something will work directly.

如果被积函数可以写成（某式）ⁿ 乘以该式的导数，那么直接用 u = 该式进行代换即可成功。

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3. Basic Substitution with Polynomials | 多项式的基本代换

egral becomes ∫ u⁴ du, which equals (1/5)u⁵ + C. Substituting back gives (1/5)(x²+3)⁵ + C.

考虑 ∫ 2x(x²+3)⁴ dx。令 u = x²+3。则 du/dx = 2x，于是 du = 2x dx。积分变为 ∫ u⁴ du，等于 (1/5)u⁵ + C。代回原变量得 (1/5)(x²+3)⁵ + C。

Always remember to replace dx with the appropriate expression in terms of du. Missing this step is a common source of error.

始终记得将 dx 用关于 du 的适当表达式替换。忽略这一步是常见的错误来源。

For definite integrals, we can either change the limits to values of u or substitute back to x before applying the original limits. The former is often quicker.

对于定积分，我们可以将积分限改为 u 的值，或者在应用原积分限之前代回 x。前者通常更快捷。

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4. Working with Definite Integrals | 处理定积分

When evaluating ∫[a,b] f(g(x))g' the limits: when x = a, u = g(a); when x = b, u = g(b). Then compute ∫[g(a),g(b)] f(u) du.

当用 u = g(x) 计算 ∫[a,b] f(g(x))g'(x) dx 时，改变积分限：当 x = a 时，u = g(a)；当 x = b 时，u = g(b)。然后计算 ∫[g(a),g(b)] f(u) du。

Example: Evaluate ∫[0,1] 2x e^x² dx. Let u = x², du = 2x dx. Limits: x=0 → u=0; x=1 → u=1. The integral becomes ∫[0,1] e^u du = [e^u] from 0 to 1 = e - 1.

示例：计算 ∫[0,1] 2x e^x² dx。令 u = x²，du = 2x dx。积分限：x=0 → u=0；x=1 → u=1。积分变为 ∫[0,1] e^u du = [e^u] 从 0 到 1 = e - 1。

Do not forget to adjust the limits when you perform the substitution; otherwise you might incorrectly use x-values in an expression in u.

进行代换时不要忘记调整积分限；否则可能会将 x 值错误地用于以 u 表示的表达式。

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5. Substitution Involving Exponentials and Logarithms | 涉及指数与对数的代换

Integrals of the form ∫ f'(x) e^(f(x)) dx are ideal for substitution. Let u = f(x), then du = f'(x) dx, so ∫ e^u du = e^u + C.

形如 ∫ f'(x) e^(f(x)) dx 的积分非常适合代换。令 u = f(x)，则 du = f'(x) dx，于是 ∫ e^u du = e^u + C。

Similarly, ∫ f'(x)/f(x) dx yields ln|f(x)| + C. This can be proved by letting u = f(x), giving ∫ (1/u) du.

类似地，∫ f'(x)/f(x) dx 得到 ln|f(x)| + C。这可以通过令 u = f(x) 得到 ∫ (1/u) du 来证明。

For example, ∫ (3x²+2)/(x³+2x) dx: set u = x³+2x, du = (3x²+2) dx, integral = ln|u| + C = ln|x³+2x| + C.

例如，∫ (3x²+2)/(x³+2x) dx：设 u = x³+2x，du = (3x²+2) dx，积分 = ln|u| + C = ln|x³+2x| + C。

• Always check that the numerator is exactly the derivative of the denominator.
• 始终检查分子是否恰好是分母的导数。

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6. Trigonometric Substitutions | 三角代换

et u = sin x, du = cos x dx. The integral becomes ∫ u³ du = (1/4)u⁴ + C = (1/4) sin⁴x + C.

在积分正弦和余弦的幂时，代换可以简化积分。例如，∫ sin³x cos x dx：令 u = sin x，du = cos x dx。积分变为 ∫ u³ du = (1/4)u⁴ + C = (1/4) sin⁴x + C。

dx, rewrite as ∫ sin x / cos x dx and substitute u = cos x, du = -sin x dx, leading to -ln|cos x| + C = ln|sec x| + C.

如果遇到 ∫ tan x dx，可改写为 ∫ sin x / cos x dx，并令 u = cos x，du = -sin x dx，得到 -ln|cos x| + C = ln|sec x| + C。

more complex expressions like ∫ sin(2x) cos⁵(2x) dx can be tackled by letting u = cos(2x).

更复杂的表达式，如 ∫ sin(2x) cos⁵(2x) dx，可以通过令 u = cos(2x) 来解决。

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7. Reverse Chain Recognition | 逆向链式法则识别

With practice, you can integrate some functions directly by mentally applying the reverse of the chain rule. This is essentially substitution done swiftly.

通过练习，你可以通过心算直接运用链式法则的逆过程来积分某些函数。这本质上就是快速代换。

For example, ∫ (2x+1) e^x²+x dx: the derivative of the exponent x²+x is 2x+1, so the answer is e^x²+x + C.

例如，∫ (2x+1) e^x²+x dx：指数 x²+x 的导数是 2x+1，因此答案为 e^x²+x + C。

This method is invaluable in Edexcel exams where questions are designed to test fluency in recognising the pattern without writing full substitution steps every time.

这种方法在Edexcel考试中非常宝贵，试题旨在考察学生不用每次都写出完整代换步骤就能识别模式的能力。

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8. Handling Fractions and Rationalising Substitutions | 处理分式与有理化代换

Sometimes an integral contains a troublesome root like √ax+b. A substitution u = √ax+b or u² = ax+b can rationalise the integrand.

有时积分中含有像 √ax+b 这样的棘手根式。令 u = √ax+b 或 u² = ax+b 可以将被积函数有理化。

Example: ∫ x√x+1 dx. Let u = x+1, then x = u-1 and dx = du. The integral becomes ∫ (u-1)√u du = ∫ (u^3/2 - u^1/2) du = (2/5)u^5/2 - (2/3)u^3/2 + C. Substitute back.

示例：∫ x√x+1 dx。令 u = x+1，则 x = u-1，dx = du。积分变为 ∫ (u-1)√u du = ∫ (u^3/2 - u^1/2) du = (2/5)u^5/2 - (2/3)u^3/2 + C。然后代回。

This approach is powerful when a linear expression appears inside a radical.

当根号内出现线性表达式时，这种方法非常有效。

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9. Substitution with Inverse Trigonometric Functions | 涉及反三角函数的代换

Standard forms like ∫ 1/√a² - x² dx use a substitution x = a sin θ (or a cosθ). Then dx = a cos θ dθ, and √a² - x² becomes a cos θ, leading to ∫ 1 dθ = θ + C = arcsin(x/a) + C.

标准形式如 ∫ 1/√a² - x² dx 使用代换 x = a sin θ（或 a cosθ）。则 dx = a cos θ dθ，√a² - x² 变为 a cos θ，从而得到 ∫ 1 dθ = θ + C = arcsin(x/a) + C。

, the substitution x = a tan θ works because 1 + tan²θ = sec²θ, simplifying the integrand.

类似地，对于 ∫ 1/(a² + x²) dx，代换 x = a tan θ 有效，因为 1 + tan²θ = sec²θ，可简化被积函数。

These trigonometric substitutions are essential for Edexcel students aiming for top grades, as they frequently appear in more challenging integration problems.

这些三角代换对于志在取得高分的学生至关重要，它们经常出现在更具挑战性的积分问题中。

x = a sin θ ⇒ √a² - x² = a cos θ; x = a tan θ ⇒ a² + x² = a² sec²θ

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10. Common Pitfalls and How to Avoid Them | 常见陷阱与避免方法

Pitfall / 陷阱	Solution / 解决方法
Forgetting to convert dx to du	Write du = g'(x) dx explicitly and replace dx.
忘记将 dx 转换为 du	明确写出 du = g'(x) dx 并替换 dx。
Not adjusting limits for definite integrals	Change limits to u-values or substitute back before applying limits.
未对定积分调整积分限	将积分限改为 u 值，或在应用限值前代回原变量。
Choosing the wrong substitution	Look for an inner function whose derivative is present; if not, try to create it by multiplying constants.
选择了错误的代换	寻找其导数出现的内部函数；若没有，尝试通过乘常数来构造。
Leaving the answer in terms of u in a definite integral after reverting limits	If you substituted back to x, use original limits; if in u, use u-limits.
在代回限值后答案仍用 u 表示	若已代回 x，使用原积分限；若以 u 表示，则使用 u 限。

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11. Comparison with Integration by Parts | 与分部积分法的对比

Substitution and integration by parts tackle different kinds of products. Substitution is best when one factor is the derivative of another function; integration by parts is designed for products of two unrelated functions, like x sin x.

代换法和分部积分法处理不同种类的乘积。当其中一个因子是另一个函数的导数时，代换法最佳；分部积分法适用于两个不相关函数的乘积，如 x sin x。

Sometimes both methods are needed, but identifying the structure is key. A typical Edexcel exam question might combine a substitution with a standard form, so remain flexible.

有时两种方法都需要，但识别结构是关键。典型的Edexcel考题可能会将代换与标准形式结合，因此要保持灵活。

A quick guide: if you see e^x² * x, use substitution; if you see x e^x, use integration by parts.

快速指南：若看到 e^x² * x，用代换；若看到 x e^x，用分部积分。

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12. Exam Tips and Practice Strategy | 应试技巧与练习策略

Always show the substitution step clearly: 'Let u = ..., then du/dx = ...' to secure method marks. Even if the final answer is wrong, careful working gains credit.

始终清晰地写出代换步骤：'令 u = ...，则 du/dx = ...'，以确保获得方法分。即使最终答案错误，仔细的过程也能得分。

Practice all types: polynomial, exponential, logarithmic, trigonometric, and rationalising substitutions. Use past Edexcel papers to spot common patterns.

练习所有类型：多项式、指数、对数、三角和有理化代换。利用Edexcel历年真题发现常见模式。

After integrating, differentiate your answer to check it matches the original integrand - a brilliant way to self-verify in the exam.

积分后，对答案求导以检验其是否与原被积函数匹配——这是考试中自我验证的绝佳方法。

Finally, time yourself on substitution problems to build speed without sacrificing accuracy.

最后，对代换问题进行计时练习，提高速度而不牺牲准确性。

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