Integration by Parts: From Product Rule to Exam Technique | 分部积分法：从乘积法则到考试技巧

📚 Integration by Parts: From Product Rule to Exam Technique | 分部积分法：从乘积法则到考试技巧

Integration by parts is one of the most powerful techniques in Edexcel A-Level Pure Mathematics. It transforms the integral of a product into two simpler parts, allowing functions such as x eˣ, x sin x, and ln x to be integrated systematically. Many students can memorise the formula, but exam success depends on choosing u and dv/dx wisely, keeping signs accurate, and knowing when to apply the method repeatedly or for definite integrals.

分部积分法是 Edexcel A-Level 纯数学中最常用的技巧之一。它把一个乘积函数的积分拆成两个更简单的部分，使学生能够系统地求解 x eˣ、x sin x 和 ln x 等函数的积分。很多同学能背出公式，但考试拿高分的关键在于合理选择 u 和 dv/dx、保持符号准确，以及判断何时需要重复使用或处理定积分。

1. Why Integration by Parts Exists | 为什么需要分部积分法

The product rule for differentiation states that the derivative of uv is u’v + uv’. When we reverse this process, we obtain a method for integrating products. Without integration by parts, integrals like ∫ x cos x dx cannot be solved by simple reverse chain rule or standard substitution.

乘积法则告诉我们，uv 的导数是 u’v + uv’。当我们把这个过程反过来使用时，就得到了一种求乘积积分的方法。如果没有分部积分法，像 ∫ x cos x dx 这样的积分用简单链式法则逆运算或普通换元法是很难求出来的。

2. The Formula and Derivation | 公式与推导

Let u and v be functions of x. The product rule gives d/dx (uv) = u dv/dx + v du/dx. Integrating both sides with respect to x and rearranging leads to the standard formula:

∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx

In shorthand, we often write ∫ u dv = uv − ∫ v du. This is quoted in the Edexcel formulae booklet, but you must be able to apply it quickly and accurately.

设 u 和 v 都是 x 的函数。由乘积法则可得 d/dx (uv) = u dv/dx + v du/dx。对等式两边关于 x 积分并移项，就得到标准公式：

∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx

简写形式常记作 ∫ u dv = uv − ∫ v du。Edexcel 公式手册中会给出这个公式，但考试中你必须能够快速且准确地应用它。

3. Choosing u and dv: The LIATE Guideline | 选择 u 和 dv：LIATE 原则

A good choice of u should simplify when differentiated, while dv/dx should be something that can be integrated easily. The LIATE order can help: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Choose u as high as possible on this list.

好的 u 选择应该是在求导后变得更简单，而 dv/dx 应该是容易积分的部分。LIATE 顺序可以帮助你：对数函数、反三角函数、代数函数、三角函数、指数函数。尽量把列表中位置较靠前的函数选作 u。

• English: For ∫ x eˣ dx, choose u = x (algebraic) and dv/dx = eˣ (exponential). 中文：对于 ∫ x eˣ dx，选择 u = x（代数函数），dv/dx = eˣ（指数函数）。
• English: For ∫ x ln x dx, choose u = ln x (logarithmic) and dv/dx = x. 中文：对于 ∫ x ln x dx，选择 u = ln x（对数函数），dv/dx = x。

4. Basic Polynomial × Exponential Example | 基本多项式乘指数例题

Worked example: Find ∫ x eˣ dx. Let u = x, so du/dx = 1. Let dv/dx = eˣ, so v = eˣ. Substituting into the formula gives:

∫ x eˣ dx = x eˣ − ∫ 1 · eˣ dx = x eˣ − eˣ + C

Always add the constant of integration for indefinite integrals. The term x eˣ has been reduced to a simpler integral that can be evaluated directly.

例题：求 ∫ x eˣ dx。设 u = x，则 du/dx = 1。设 dv/dx = eˣ，则 v = eˣ。代入公式得到：

∫ x eˣ dx = x eˣ − ∫ 1 · eˣ dx = x eˣ − eˣ + C

不定积分一定要加上积分常数 C。这里的 x eˣ 被化简成了一个可以直接求出的更简单积分。

5. Logarithmic Functions | 对数函数

To integrate ln x, write it as 1 · ln x. Choose u = ln x and dv/dx = 1. Then du/dx = 1/x and v = x. Therefore:

∫ ln x dx = x ln x − ∫ x(1/x) dx = x ln x − x + C

This trick works for other logarithmic forms, such as ∫ (ln x)² dx, where repeated integration by parts is required. In Edexcel questions, logarithmic integrals often appear in mixed methods and differential equation contexts.

求 ∫ ln x dx 时，可以把它写成 1 · ln x。选择 u = ln x，dv/dx = 1。于是 du/dx = 1/x，v = x。因此：

∫ ln x dx = x ln x − ∫ x(1/x) dx = x ln x − x + C

这一技巧也适用于其他对数形式，例如 ∫ (ln x)² dx，此时需要反复使用分部积分法。在 Edexcel 试题中，对数积分常出现在综合题和微分方程情境

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