Integration by Parts for Edexcel A-Level Maths | Edexcel A-Level 数学：分部积分法

📚 Integration by Parts for Edexcel A-Level Maths | Edexcel A-Level 数学：分部积分法

In Edexcel A-Level Mathematics, integration by parts is one of the core methods for evaluating integrals that are products of two fundamentally different types of functions. It is especially important for Pure Mathematics Paper 1 and Paper 2, where candidates must be able to identify when and how to apply the formula, select suitable u and dv, and handle repeated or cyclic cases efficiently.

在 Edexcel A-Level 数学中，分部积分法是计算两类不同函数乘积积分的重要方法之一。它在 Pure Mathematics Paper 1 和 Paper 2 中尤其常见，考生需要能够判断何时以及如何应用公式、选择合适的 u 和 dv，并高效处理重复或循环情形。

1. The Idea of Reverse Product Rule | 分部积分法的原理：乘积法则的逆运算

Integration by parts comes directly from the product rule for differentiation. If u and v are functions of x, then (uv)’ = u’v + uv’. Rearranging gives uv’ = (uv)’ − u’v. Integrating both sides with respect to x produces the standard formula.

分部积分法直接源自乘积法则。若 u 和 v 都是 x 的函数，则 (uv)’ = u’v + uv’。移项可得 uv’ = (uv)’ − u’v。对两边关于 x 积分，就得到标准公式。

∫ u dv = uv − ∫ v du

In this formula, u is usually chosen as the part that simplifies when differentiated, while dv is chosen as the part that can be integrated easily. The differential form uses du = u’ dx and v = ∫ dv.

在这个公式中，通常选择 u 为求导后变得更简单的部分，而 dv 选择容易积分的部分。微分形式使用 du = u’ dx 和 v = ∫ dv。

2. Choosing u: LIATE Rule | 选择 u：LIATE 法则

A common heuristic for choosing u is the LIATE rule, which gives priority in descending order:

选择 u 的一个常用启发式方法是 LIATE 法则，按优先级从高到低排列：

• L — Logarithmic functions such as ln x, logₐx | 对数函数，如 ln x、logₐx
• I — Inverse trigonometric functions such as arctan x, arcsin x | 反三角函数，如 arctan x、arcsin x
• A — Algebraic functions such as xⁿ, polynomials | 代数函数，如 xⁿ、多项式
• T — Trigonometric functions such as sin x, cos x | 三角函数，如 sin x、cos x
• E — Exponential functions such as eˣ, aˣ | 指数函数，如 eˣ、aˣ

However, LIATE is only a guideline. In Edexcel exam questions, you should always think about what makes the new integral ∫ v du simpler than the original. Sometimes algebraic × exponential uses u = algebraic, but algebraic × trigonometric may also use tabular integration.

不过 LIATE 只是一个指引。在 Edexcel 考题中，你始终要考虑如何让新积分 ∫ v du 比原积分更简单。有时代数函数 × 指数函数取 u 为代数函数；而代数函数 × 三角函数也可用表格法积分。

3. Basic Example: ∫ x eˣ dx | 基础例题：∫ x eˣ dx

Evaluate ∫ x eˣ dx. By LIATE, choose u = x (algebraic) and dv = eˣ dx (exponential). Then du = dx and v = eˣ.

计算 ∫ x eˣ dx。根据 LIATE，选择 u = x（代数函数），dv = eˣ dx（指数函数）。则 du = dx，v = eˣ。

• u = x, du = dx | u = x，du = dx
• dv = eˣ dx, v = eˣ | dv = eˣ dx，v = eˣ

Substituting into the integration by parts formula gives:

代入分部积分公式得到：

∫ x eˣ dx = x eˣ − ∫ eˣ dx

The remaining integral is straightforward: ∫ eˣ dx = eˣ + C. Therefore,

剩下的积分很简单：∫ eˣ dx = eˣ + C。因此，

∫ x eˣ dx = x eˣ − eˣ + C

Always include the constant of integration for indefinite integrals. A common Edexcel mark is awarded for the final +C, and omitting it can cost accuracy marks.

不定积分必须加上积分常数。Edexcel 通常会为最后的 +C 给

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