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

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📚 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)。