📚 Mastering Advanced Integration: Substitution and Parts for Edexcel A-Level | 掌握高级积分法：Edexcel A-Level 的换元与分部积分
Integration is one of the most demanding topics in Edexcel A-Level Pure Mathematics. In particular, integration by substitution and integration by parts appear regularly in Paper 1 and Paper 2, often within multi-step problems involving trigonometric, exponential and logarithmic functions. This article explains the core methods, common pitfalls and exam-style strategies needed to score full marks.
积分是 Edexcel A-Level 纯数学中要求最高的主题之一。尤其是换元积分法与分部积分法经常出现在 Paper 1 和 Paper 2 中，并且常常出现在涉及三角函数、指数函数和对数函数的多步问题里。本文将讲解核心方法、常见错误以及考试中取得满分所需的策略。
1. The Edexcel Exam Context | Edexcel 考试背景
In Edexcel A-Level Mathematics, advanced integration appears in Pure Mathematics Papers 1 and 2, and it underpins many applied problems in mechanics and statistics. You are expected to integrate functions containing exponential, logarithmic, trigonometric and algebraic combinations accurately and efficiently.
在 Edexcel A-Level 数学中，高级积分出现在纯数学 Paper 1 和 Paper 2 中，并且是力学与统计学许多应用问题的基础。考试要求你能够准确、高效地对含有指数函数、对数函数、三角函数和代数组合的函数进行积分。
Two techniques dominate the harder questions: integration by substitution and integration by parts. They are listed in the Edexcel formula booklet, but using them correctly requires judgement and practice.
换元积分法和分部积分法是较难考题中的两大核心技巧。它们虽然列在 Edexcel 公式手册中，但正确运用需要判断力和大量练习。
2. Reverse Chain Rule and Substitution | 逆链式法则与换元法
For an expression of the form ∫ f(g(x)) g'(x) dx, the substitution u = g(x) turns the integral into ∫ f(u) du. This is the reverse of the chain rule.
对于形如 ∫ f(g(x)) g'(x) dx 的表达式，令 u = g(x) 可把积分化为 ∫ f(u) du。这正是链式法则的逆过程。
∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x)
其中 u = g(x)。
A typical case is ∫ 2x (x² + 1)⁵ dx. Let u = x² + 1, so du/dx = 2x and dx = du/(2x). The integral becomes ∫ u⁵ du = u⁶/6 + C, giving (x² + 1)⁶/6 + C after substituting back.
一个典型案例是 ∫ 2x (x² + 1)⁵ dx。令 u = x² + 1，则 du/dx = 2x，dx = du/(2x)。积分变为 ∫ u⁵ du = u⁶/6 + C，代回后得到 (x² + 1)⁶/6 + C。
3. Definite Integrals and Changing Limits | 定积分与上下限转换
For definite integrals, you must change the limits from x-values to u-values before evaluating, or convert back to x after integration. Changing limits usually saves time and reduces sign errors.
对于定积分，必须在积分前将上下限从 x 值转换为 u 值，或者在积分后换回 x 再代入。通常转换上下限能节省时间并减少符号错误。
Consider ∫01 2x (x² + 1)⁵ dx. When x = 0, u = 1; when x = 1, u = 2. The integral becomes ∫12 u⁵ du.
考虑 ∫01 2x (x² + 1)⁵ dx。当 x = 0 时，u = 1；当 x = 1 时，u = 2。积分变为 ∫12 u⁵ du。
∫12 u⁵ du = [ u⁶/6 ]12 = 64/6 − 1/6 = 63/6 = 21/2
4. Integration by Parts Formula | 分部积分公式
Integration by parts is derived from the product rule. The formula is ∫ u dv = uv − ∫ v du. In Edexcel notation, we often write ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx.
分部积分法由乘法法则推导而来。公式为 ∫ u dv = uv − ∫ v du。在 Edexcel 记号中，常写作 ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx。
∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx
The key is to choose u as the part that becomes simpler when differentiated, and dv/dx as the part that is easy to integrate.
关键在于选择微分后会变简单的部分作为 u，而将容易积分的部分作为 dv/dx。
5. Choosing u and dv/dx with LIATE | 用 LIATE 原则选择 u 与 dv/dx
A useful priority order for choosing u is LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Choose u from the earliest category in this list.
选择 u 的一个实用优先顺序是 LIATE：对数函数、反三角函数、代数函数、三角函数、指数函数。从列表中较早出现的类别中选取 u。
For ∫ x eˣ dx, choose u = x and dv/dx = eˣ. Then du/dx = 1 and v = eˣ, giving ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.
对于 ∫ x eˣ dx，选择 u = x、dv/dx = eˣ。则 du/dx = 1、v = eˣ，得到 ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。
For ∫ x ln x dx, choose u = ln x because logarithmic has higher priority than algebraic. Then dv/dx = x, so du/dx = 1/x and v = ½ x².
对于 ∫ x ln x dx，选择 u = ln x，因为对数函数在优先顺序中早于代数函数。然后令 dv/dx = x，则 du/dx = 1/x、v = ½ x²。
6. Repeated Integration by Parts | 重复分部积分
When the remaining integral still contains a product, apply integration by parts again. This often happens with ∫ x² eˣ dx or ∫ x² sin x dx.
当剩余的积分仍含有乘积时，需要再次使用分部积分。常见于 ∫ x² eˣ dx 或 ∫ x² sin x dx。
For ∫ x² eˣ dx, first choose u = x², dv/dx = eˣ. This gives x² eˣ − ∫ 2x eˣ dx. Then apply parts again to ∫ 2x eˣ dx.
对于 ∫ x² eˣ dx，先选择 u = x²、dv/dx = eˣ。得到 x² eˣ − ∫ 2x eˣ dx。然后对 ∫ 2x eˣ dx 再次使用分部积分。
∫ x² eˣ dx = x² eˣ − 2(x eˣ − eˣ) + C = eˣ(x² − 2x + 2) + C
7. Special Cases: ln x and eˣ sin x | 特殊情况：ln x 与 eˣ sin x
To integrate ln x, write it as ∫ ln x · 1 dx and choose u = ln x, dv/dx = 1. This gives ∫ ln x dx = x ln x − x + C.
积分 ln x 时，写成 ∫ ln x · 1 dx 并选择 u = ln x、dv/dx = 1。可得 ∫ ln x dx = x ln x − x + C。
For ∫ eˣ sin x dx, apply parts twice and rearrange. The same integral reappears, allowing you to solve for it algebraically. This cyclic pattern is common in Edexcel exam questions.
对于 ∫ eˣ sin x dx，需要两次分部积分并移项。积分会重新出现，从而可以通过代数方法求解。这种循环模式在 Edexcel 考试题中很常见。
8. Using Standard Results and Identities | 使用标准结果与三角恒等式
Before integrating, simplify using trigonometric identities such as sin² x = ½(1 − cos 2x) and cos² x = ½(1 + cos 2x). These turn squared trig functions into integrable linear terms.
积分前先用三角恒等式化简，例如 sin² x = ½(1 − cos 2x) 和 cos² x = ½(1 + cos 2x)。这些恒等式将二次三角函数化为可积分的线性项。
Also recognise standard Edexcel results: ∫ sec² x dx = tan x + C, ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C.
同时要熟悉 Edexcel 的标准结果：∫ sec² x dx = tan x + C，∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C。
9. Common Mistakes in the Exam | 考试常见错误
The most common errors are forgetting to change the limits in a definite substitution, writing du instead of dx incorrectly, dropping the constant of integration, and choosing u and dv/dx the wrong way round.
最常见错误包括：定积分换元时忘记转换上下限、错误处理 du 与 dx、漏掉积分常数 C，以及 u 和 dv/dx 选择颠倒。
Another serious error is mixing up ½ and 2 when using double-angle identities. Always check by differentiation after integrating.
另一个严重错误是在使用倍角恒等式时把 ½ 和 2 搞混。积分后务必通过求导进行检验。
10. Worked Exam-Style Example | 考试型例题精讲
Question: Evaluate ∫ from x = 0 to x = π/4 of x sin 2x dx.
题目：计算 ∫ 从 0 到 π/4 的 x sin 2x dx。
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