📚 Integration by Substitution (Edexcel A-Level) | 换元积分法（爱德思A-Level）
Integration by substitution is one of the most powerful techniques in A-Level calculus. It allows you to transform complicated integrals into standard forms that are easy to evaluate. In the Edexcel specification, substitution (often called the reverse chain rule) appears frequently in Pure Mathematics 3 and 4, combined with trigonometric, exponential, and rational functions. Mastering this method is essential for tackling exam questions involving composite functions and definite integrals with limits.
换元积分法是A-Level微积分中最强大的技巧之一。它能将复杂的积分转化为易于计算的标准形式。在爱德思考试大纲中，换元法（常被称为反向链式法则）频繁出现在P3和P4试卷中，与三角函数、指数函数和有理函数结合。掌握这一方法对于解决复合函数的积分以及带限的定积分问题至关重要。
1. The Core Idea: Reverse of the Chain Rule | 核心思想：链式法则的逆过程
tion reverses the chain rule for differentiation. If y = f(u) and u = g(x), then dy/dx = f'(u) · g'(x). When integrating, you often see an expression that looks like f'(g(x)) · g'(x). By letting u = g(x), you simplify the integral to ∫ f'(u) du, which is usually straightforward.
从本质上讲，换元积分法是微分链式法则的逆过程。若 y = f(u) 且 u = g(x)，则 dy/dx = f'(u) · g'(x)。积分时，你常常会看到形如 f'(g(x)) · g'(x) 的表达式。通过令 u = g(x)，就可将积分简化为 ∫ f'(u) du，这通常很容易计算。
then du/dx = 2x, so du = 2x dx. The integral becomes ∫ cos u du = sin u + C = sin(x²) + C.
例如，考虑 ∫ 2x cos(x²) dx。注意 x² 的导数是 2x，正好作为一个因子出现。令 u = x²，则 du/dx = 2x，因此 du = 2x dx。积分变为 ∫ cos u du = sin u + C = sin(x²) + C。
2. Formal Substitution Procedure | 正式的换元步骤
The formal method involves choosing a suitable u, calculating du/dx, and rewriting the integral entirely in terms of u and du. For definite integrals you also change the limits. The goal is to eliminate all x terms and leave a simpler integral in u.
正式方法包括选择一个合适的 u，计算 du/dx，然后将整个积分用 u 和 du 重写。对于定积分，还需更换积分限。目标是消去所有 x 项，留下一个关于 u 的更简单的积分。
tive also appears (or can be adjusted by a constant). (ii) Set u = g(x). (iii) Find du = g' e integrals, replace u with g(x); for definite integrals, evaluate using the new u-limits.
步骤解析：(i) 找出一个 x 的函数，其导数也出现（或可通过常数倍调整）。(ii) 设 u = g(x)。(iii) 求得 du = g'(x) dx。(iv) 替换所有 x 表达式和 dx。(v) 对 u 积分。(vi) 对于不定积分，将 u 代回 g(x)；对于定积分，用新的 u 限值计算。
3. Spotting the Reverse Chain Rule Pattern | 识别反向链式法则的模式
To apply substitution effectively, you must learn to spot the pattern f'(g(x)) · g' vative is not an exact match, but you can introduce a constant factor. For instance, ∫ x ex² dx can be handled by u = x², noting that x dx = (1/2) du.
要有效应用换元法，你必须学会识别模式 f'(g(x)) · g'(x)。有时导数并非精确匹配，但你可以引入常数因子。例如，∫ x ex² dx 可通过令 u = x² 处理，注意到 x dx = (1/2) du。
rks because du = cos x dx, transforming the integral to ∫ u³ du. The key is to identify an 'inner function' whose derivative (up to a constant) is sitting nearby.
更复杂的例子包括 ∫ sin³x cos x dx，其中 u = sin x 可行，因为 du = cos x dx，将积分转化为 ∫ u³ du。关键在于找出一个“内层函数”，其导数（相差一个常数倍）就在旁边。
4. Guidelines for Choosing u | 选择 u 的指南
Selecting u is often the trickiest part. General guidelines include: (a) Let u be the 'inside' of a composite function, such as the exponent in e5x² or the argument of sin. (b) Let u be a denominator or a radical, like √x+1 integrand is a rational function of (ax+b), e.g. ∫ (2x+1)/(x²+x+3) dx suggests u = x²+x+3.
选择 u 通常是最棘手的部分。一般准则包括：(a) 令 u 为复合函数的“内部”，如 e5x² 中的指数或 sin 的参数。(b) 令 u 为分母或根式，如 √x+1。(c) 当被积函数是 (ax+b) 的有理函数时，令 u 为线性表达式，例如 ∫ (2x+1)/(x²+x+3) dx 可设 u = x²+x+3。
Avoid letting u be something that makes the integral more complicated. In exams, Edexcel often provides the substitution for harder problems, but for standard ones you must choose wisely. Practice is essential.
避免设 u 使得积分更复杂。在考试中，爱德思常为较难题目给出换元，但对于标准积分，你需明智选择。练习至关重要。
5. Standard Substitutions and a Handy Table | 标准换元与速查表
cos power is odd), then use sin²x + cos²x = 1. For even powers, double-angle identities may be needed first. Another standard substitution is u = tan x for integrals involving sec²x.
三角积分常需巧妙的换元。对于 ∫ sinᵐx cosⁿx dx，若某个幂次是奇数，则令 u = cos x（若 sin 奇次）或 u = sin x（若 cos 奇次），然后利用 sin²x + cos²x = 1。若幂次均为偶数，可能需先用二倍角公式再换元。另一标准换元是 u = tan x，用于含有 sec²x 的积分。
The table below summarises common integral forms and their standard substitutions, which are extremely useful in Edexcel P3/P4.
下表总结了常见的积分形式及其标准换元，这些在爱德思 P3/P4 中极为有用。
| Integral Form | Substitution | Why It Works |
|---|---|---|
| ∫ f(ax + b) dx | u = ax + b | du = a dx, so dx = du/a |
| ∫ g'(x) / g(x) dx | u = g(x) | du = g'(x) dx, gives ln|u| |
| ∫ sinᵐx cosⁿx dx (m odd) | u = cos x | Saves one sin x for -du |
| ∫ sinᵐx cosⁿx dx (n odd) | u = sin x | Saves one cos x for du |
| ∫ √a² − x² dx | x = a sin θ | dx = a cos θ dθ, √ becomes a cos θ |
| ∫ √a² + x² dx | x = a tan θ | dx = a sec²θ dθ |
| ∫ √x² − a² dx | x = a sec θ | dx = a sec θ tan θ dθ |
6. Trig Substitutions for Quadratic Radicals | 针对二次根式的三角换元
When an integral contains √a² − x², √a² + x², or √x² − a², the standard substitutions are
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