📚 Mastering Integration by Substitution (Edexcel A-Level Maths) | 掌握换元积分法（Edexcel A-Level 数学）
In Edexcel A-Level Mathematics, integration by substitution is one of the most powerful techniques for evaluating integrals that are not immediately standard. It reverses the chain rule and enables students to handle composite functions, trigonometric integrals, and expressions involving square roots or exponentials. This article covers the method, standard substitutions, worked examples, and exam strategies.
在 Edexcel A-Level 数学中，换元积分法是求解非标准积分最有力的方法之一。它反用链式法则，使学生能够处理复合函数、三角积分以及含有根号或指数的表达式。本文涵盖换元方法、标准换元形式、例题和考试策略。
1. The Chain Rule in Reverse | 反用链式法则
Differentiation in Edexcel Pure Mathematics relies heavily on the chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). Integration by substitution reverses this process. When an integrand contains a composite function multiplied by the derivative of the inner function, we can choose u = g(x) and rewrite the integral in terms of u. This turns a complicated integral into a standard one.
Edexcel 纯数学中的微分大量依赖链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。换元积分法反用这一过程。当被积函数含有复合函数乘以内层函数的导数时，可令 u = g(x)，将积分改写为关于 u 的积分，从而把复杂积分化为标准积分。
∫ f'(g(x))g'(x) dx = f(g(x)) + C
∫ f'(u) du = f(u) + C, where u = g(x)
This is why substitution is often called the ‘anti-chain rule’ method. In exam questions, you may be given the substitution or asked to choose one yourself. Edexcel often tests the skill of spotting the inner function and rewriting the differential correctly.
因此换元法常被称为“反链式法则”方法。考试题可能给出换元式，也可能要求你自己选择。Edexcel 经常考查识别内层函数以及正确改写微分的能力。
2. Recognising When to Use Substitution | 识别何时使用换元法
Use substitution when the integrand is a product of a composite function and the derivative of its inner function, a quotient whose numerator is closely related to the derivative of the denominator, or an expression containing a linear function inside a power or root. Another clue is when integration by inspection is not immediate but one part of the integrand is the derivative of another part.
当被积函数是复合函数与其内层函数导数的乘积、分子与分母的导数密切相关，或者幂或根号内含有线性函数时，应考虑换元法。另一个线索是：无法直接观察积分，但被积函数的一部分是另一部分的导数。
Common patterns in Edexcel questions include:
∫ x e^(x²) dx ; inner function x², derivative 2x present up to constant.
∫ (2x+1)/(x²+x+1) dx ; derivative of denominator is 2x+1.
∫ √(2x+3) dx ; linear expression inside root.
∫ cos(x³) x² dx ; derivative of x³ is 3x².
Edexcel 题目中的常见模式包括：
∫ x e^(x²) dx；内层函数为 x²，其导数 2x 仅相差常数倍。
∫ (2x+1)/(x²+x+1) dx；分母的导数为 2x+1。
∫ √(2x+3) dx；根号内是线性表达式。
∫ cos(x³) x² dx；x³ 的导数为 3x²。
3. The Basic u-Substitution Procedure | 基本 u 换元步骤
Follow these steps when evaluating an indefinite integral by substitution. Step 1: Choose u = g(x), often the inner function, denominator, or expression under a root. Step 2: Differentiate to find du/dx, then write du = g'(x) dx. Step 3: Express all remaining x terms in terms of u; this may require rearranging. Step 4: Evaluate the new integral in u. Step 5: Substitute back u = g(x) for indefinite integrals.
用换元法求不定积分时，遵循以下步骤。第一步：选择 u = g(x)，通常是内层函数、分母或根号内的表达式。第二步：求导得到 du/dx，然后写出 du = g'(x) dx。第三步：将所有剩余的 x 项用 u 表示；这可能需要代数变形。第四步：求出关于 u 的新积分。第五步：对不定积分代回 u = g(x)。
For definite integrals, an important modification is to convert the limits at the same time. This is treated in the next section. In all cases, the final answer should not contain both x and u, except during intermediate working.
对于定积分，一个重要的调整是同时更换上下限。这一点将在下一节讨论。无论哪种情况，最终答案不应同时含有 x 和 u，中间步骤除外。
4. Definite Integrals and Changing Limits | 定积分与更换上下限
For a definite integral, after choosing u = g(x), convert the limits: when x = a, u = g(a); when x = b, u = g(b). Then integrate with respect to u using the new limits. This avoids back substitution and reduces operational errors, especially in exam conditions.
对于定积分，在选定 u = g(x) 后，应更换上下限：当 x = a 时，u = g(a)；当 x = b 时，u = g(b)。然后使用新上下限对 u 积分。这样可以避免回代，减少考试中的操作失误。
∫ₐᵇ f(g(x))g'(x) dx = ∫ from u=g(a) to u=g(b) f(u) du
It is essential to write the new limits clearly. Edexcel mark schemes often award marks for changing limits correctly even if the later integration contains a minor slip. Never leave the original x limits unchanged when evaluating a definite integral in u.
清晰写出新上下限至关重要。Edexcel 阅卷方案通常即使后续积分出现小错，也会对正确更换上下限给分。在涉及 u 的定积分中，绝不能保留原来的 x 上下限。
5. Standard Substitutions | 标准换元形式
The table below summarises substitutions frequently used in Edexcel Pure Mathematics. Recognising these forms allows you to choose u efficiently instead of trial and error.
下表总结了 Edexcel 纯数学中常用的换元形式。识别这些形式可以让你高效选择 u，而不是反复试错。
| Integral form | Suggested substitution |
| ∫ f(ax+b) dx | u = ax+b |
| ∫ x e^(x²) dx | u = x² |
| ∫ (2x)/(x²+1) dx | u = x²+1 |
| ∫ sinⁿx cos x dx | u = sin x |
| ∫ 1/√(a²-x²) dx | x = a sin θ |
| ∫ 1/(x²+a²) dx | x = a tan θ |
Note that in the last two rows the substitution is of the form x = … rather than u = g(x). This is common in trigonometric substitution, where the variable of integration changes from x to θ.
注意，最后两行的换元形式是 x = …，而不是 u = g(x)。这在三角换元中很常见，积分变量从 x 变为 θ。
6. Trigonometric Substitutions | 三角换元
When the integrand
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