📚 Integration by Substitution for Edexcel A Level Pure Maths | Edexcel A Level 纯数学：换元积分法
Integration by substitution is a core technique in the Edexcel A Level Pure Mathematics syllabus. It reverses the chain rule and allows you to integrate composite functions by changing the variable. Mastering this method is essential for questions on calculus, areas under curves and differential equations in both AS and A Level papers.
换元积分法是 Edexcel A Level 纯数学课程中的核心技巧。它逆转链式法则，通过更换变量来积分复合函数。掌握这一方法对于 AS 和 A Level 试卷中的微积分、曲线下面积以及微分方程题目都至关重要。
1. What Makes Substitution Necessary? | 为什么需要换元积分法
Not every integral can be evaluated by direct reverse differentiation. When an integrand contains a composite function multiplied by the derivative of the inner function, substitution is the most reliable method.
并非每个积分都能直接通过逆微分求出。当被积函数包含复合函数并乘以内层函数的导数时，换元法是最可靠的方法。
For example, the integral ∫ 2x(x² + 1)³ dx is not immediate, but the factor 2x is exactly the derivative of the inner function x² + 1. Substitution turns the expression into a simple power integral.
例如，积分 ∫ 2x(x² + 1)³ dx 不能直接得到，但因子 2x 恰好是内层函数 x² + 1 的导数。换元可将其化为简单的幂函数积分。
In Edexcel exams, substitution questions often appear without explicit instructions on what to use, so recognising this pattern is a key skill.
在 Edexcel 考试中，换元题经常不会明确告诉你该用什么代换，因此识别这一模式是一项关键能力。
2. The Reverse Chain Rule Idea | 反链式法则思想
Differentiation uses the chain rule: if y = f(g(x)), then dy/dx = f'(g(x))g'(x). Integration by substitution reverses this process by letting u = g(x), so du = g'(x) dx.
微分使用链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x))g'(x)。换元积分恰好逆转这一过程，令 u = g(x)，于是 du = g'(x) dx。
The integral ∫ f'(g(x))g'(x) dx becomes ∫ f'(u) du = f(u) + C. This identity is the foundation of the method.
积分 ∫ f'(g(x))g'(x) dx 变为 ∫ f'(u) du = f(u) + C。这个恒等式正是换元积分法的基础。
Always write down du/dx before changing the integral, because this single line prevents most algebraic slips in later steps.
在改写积分之前一定要先写出 du/dx，因为这一行可以防止后续步骤中大部分代数错误。
3. Choosing u: General Strategy | 选择 u 的一般策略
Choosing the right substitution is the most important skill. In Edexcel questions, u is often given, but when it is not, look for an inner function whose derivative appears in the integrand.
选择正确的换元最关键。在 Edexcel 试题中，u 通常已给出；若未给出，则寻找其导数出现在被积函数中的内层函数。
The following table summarises common choices of u for typical integrand forms:
下表总结了典型被积函数形式下常见的 u 的选择：
| Integrand form | Choose u as |
|---|---|
| Function inside a bracket, root or denominator | The inner function |
| f(ax + b) | u = ax + b |
| f(x² + a) multiplied by x | u = x² + a |
| f(sin x) multiplied by cos x | u = sin x |
| f(cos x) multiplied by sin x | u = cos x |
| f(e^x) multiplied by e^x | u = e^x |
| f(ln x) multiplied by 1/x | u = ln x |
If a question gives you a specific substitution, use it exactly as stated, even if another route seems possible.
如果题目给出了指定的代换，请严格按照题目的要求使用，即使看起来还有其他方法。
4. Worked Example: Polynomial Composite | 例题：
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