📚 Edexcel A-Level Pure Mathematics: Integration by Substitution | Edexcel A-Level 纯数学：换元积分法
Integration by substitution is one of the most powerful techniques in the Edexcel A-Level Pure Mathematics specification. It reverses the chain rule and allows you to integrate composite functions by changing the variable to simplify the integral. Mastery of this method is essential for both Paper 1 and Paper 2, especially for questions involving powers, exponentials, logarithms, and trigonometric functions.
换元积分法是 Edexcel A-Level 纯数学大纲中最重要的技巧之一。它反转链式法则，通过更换变量来简化复合函数的积分。掌握这一方法对 Paper 1 和 Paper 2 都至关重要，尤其是涉及幂函数、指数函数、对数函数和三角函数的题目。
1. What Is Integration by Substitution? | 什么是换元积分法？
Integration by substitution is a technique for evaluating integrals of the form ∫ f(g(x))g'(x) dx. By letting u = g(x), the differential du is equal to g'(x)dx, so the integral becomes ∫ f(u) du. This is often much easier to integrate than the original expression.
换元积分法用于求解形如 ∫ f(g(x))g'(x) dx 的积分。令 u = g(x)，则微分 du = g'(x)dx，因此积分变为 ∫ f(u) du。这通常比原式更容易积分。
The method works because integration reverses differentiation. If differentiating f(u) with respect to x gives f'(u)u’ by the chain rule, then integrating f'(u)u’ with respect to x must return f(u) + C.
该方法之所以有效，是因为积分是微分的逆过程。如果对 x 求导 f(u) 得到 f'(u)u’（链式法则），那么对 x 积分 f'(u)u’ 必定得到 f(u) + C。
∫ f(g(x))g'(x) dx = ∫ f(u) du, where u = g(x)
2. The Reverse Chain Rule Connection | 与逆链式法则的联系
The reverse chain rule is a special case of integration by substitution. If an integrand contains a function and its derivative multiplied together, such as f'(g(x))g'(x), then the integral can be written directly as f(g(x)) + C. Recognising these patterns saves time in exams.
逆链式法则是换元积分法的一种特殊情况。如果被积函数包含一个函数与其导数相乘，例如 f'(g(x))g'(x)，那么积分可以直接写成 f(g(x)) + C。在考试中识别这些模式可以节省时间。
For example, the derivative of (2x+1)⁹ is 18(2x+1)⁸, so integrating 18(2x+1)⁸ gives (2x+1)⁹ + C. The constant factor 18 must be adjusted when the integrand is not an exact derivative.
例如，(2x+1)⁹ 的导数是 18(2x+1)⁸，所以对 18(2x+1)⁸ 积分得到 (2x+1)⁹ + C。当被积函数不是精确导数时，必须调整常数因子 18。
| Standard form | Integrated result |
|---|---|
| ∫ (ax+b)ⁿ dx, n ≠ -1 | (1/a) (ax+b)ⁿ⁺¹ / (n+1) + C |
| ∫ exp(ax+b) dx | (1/a) exp(ax+b) + C |
| ∫ cos(ax+b) dx | (1/a) sin(ax+b) + C |
| ∫ sin(ax+b) dx | −(1/a) cos(ax+b) + C |
| ∫ f'(x)/f(x) dx | ln|f(x)| + C |
3. Choosing the Substitution u | 如何选择代换变量 u
Choosing the right substitution is the key skill in this topic. In most A-Level questions, the substitution is either given in the question or indicated by the structure of the integrand. You should look for an inner function whose derivative also appears approximately in the integral.
选择合适的代换变量是本主题的关键技能。在大多数 A-Level 题目中，代换变量要么在题中给出，要么由被积函数的结构提示。你应该寻找一个内层函数，其导数也近似地出现在积分中。
A useful strategy is to let u be the expression inside a bracket, under a square root, in a denominator, or in an exponent. Then check whether du/dx is present as a factor. If there is a constant factor missing, you can usually adjust for it.
一个有用的策略是令 u 为括号内、根号下、分母中或指数中的表达式。然后检查 du/dx 是否作为因子出现。如果缺少常数因子，通常可以进行调节。
For rational functions, if the numerator is a multiple of the derivative of the denominator, choose u as the denominator. This leads directly to a natural logarithm result.
对于有理函数，如果分子是分母导数的倍数，选择分母作为 u。这样直接得到自然对数的结果。
4. Worked Example: Power of a Linear Function | 例题：线性函数的幂
Worked example: Find ∫ (3x+5)⁸ dx.
例题：求
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