📚 Integration by Substitution and By Parts for Edexcel A Level Maths | Edexcel A Level 数学积分技巧：换元法与分部积分法
This revision guide focuses on two powerful techniques in Edexcel A Level Mathematics: integration by substitution and integration by parts. Both appear regularly in Pure Mathematics Paper 1 and Paper 2, especially in longer structured questions.
本复习指南聚焦 Edexcel A Level 数学中两大重要技巧：换元积分法与分部积分法。这两种方法经常出现在纯数学 Paper 1 和 Paper 2 的较难综合题中。
You are expected to recognise when a standard integral is not available and to choose the most efficient method for the job.
考试要求你能够判断标准积分公式无法直接使用的情况，并选择最有效的方法完成积分。
1. The Need for Advanced Techniques | 为什么需要高级积分技巧
Standard results only integrate simple powers of x, exponentials, sine, cosine and their obvious sums.
标准积分公式只能处理简单的 x 的幂、指数函数、正弦、余弦以及它们的线性组合。
Expressions such as x·e^(2x), x²·sin x, or √(2x+1) cannot be integrated by reversing basic differentiation alone.
像 x·e^(2x)、x²·sin x 或 √(2x+1) 这类表达式，仅靠基本微分逆运算无法积分。
Substitution targets composite functions, while integration by parts targets products of different function types.
换元法针对复合函数，分部积分法针对不同类型函数的乘积。
2. Integration by Substitution: Core Idea | 换元积分法：核心思想
If you spot a function and its derivative together, substitution can simplify the integral.
如果你在积分中看到一个函数与其导数同时出现，换元法可以简化积分。
The reverse chain rule states:
逆链式法则可以表示为：
∫ f(g(x))·g'(x) dx = ∫ f(u) du
where u = g(x) and du = g'(x) dx.
其中令 u = g(x)，则 du = g'(x) dx。
After converting every x term to u, integrate with respect to u, then substitute back.
将所有含 x 的项都转换为 u 后，对 u 积分，最后再把 u 代回 x。
3. How to Apply Substitution Step by Step | 换元法步骤详解
- Choose u = inner function or the part whose derivative appears.
- 选出 u 等于内层函数或导数已出现的部分。
- Differentiate u to find du/dx, then write dx = du / (du/dx).
- 对 u 求导得到 du/dx，然后写成 dx = du / (du/dx)。
- Replace all x expressions with u. If any x remains, reuse the substitution to eliminate it.
- 将所有含 x 的表达式替换为 u。若仍有 x 残留，再次利用换元关系消去。
- Integrate with respect to u and finally substitute back u = g(x).
- 对 u 积分，最后代回 u = g(x)。
For definite integrals, you must also change the limits from x-values to u-values.
对于定积分，还必须把积分限从 x 的取值改成 u 的取值。
4. Worked Example: Substitution | 换元法例题
Find ∫ x√(2x+1) dx using the substitution u = 2x+1.
用换元 u = 2x+1 求 ∫ x√(2x+1) dx。
Since u = 2x+1, differentiating gives du/dx = 2, so dx = du/2.
因为 u = 2x+1，求导得 du/dx = 2，所以 dx = du/2。
Also x = (u-1)/2, so the integral becomes:
同时 x = (u-1)/2，因此积分变为：
∫ x√(2x+1) dx = ∫ [(u-1)/2]√u · (du/2) = (1/4) ∫ (u^(3/2) – u^(1/2)) du
Now integrate term by term:
现在逐项积分：
= (1/4) [ (2/5)u^(5/2) – (2/3)u^(3/2) ] + C = (1/10)(2x+1)^(5/2) – (1/6)(2x+1)^(3/2) + C
This is a typical Edexcel question where you must rewrite x in terms of u.
这是 Edexcel 典型考题，需要你将 x 用 u 表示出来。
5. Def
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