Edexcel A-Level Maths: Partial Fractions & Integration – Practice Set 137 | 爱德思A-Level数学：部分分式与积分——练习集137

📚 Edexcel A-Level Maths: Partial Fractions & Integration – Practice Set 137 | 爱德思A-Level数学：部分分式与积分——练习集137

This practice set from PDF Joiner (4), page 137, focuses on one of the highest-yield topics in the Edexcel A-Level Mathematics specification: decomposing rational functions into partial fractions and then integrating the result. The techniques covered here appear regularly in Pure 3 and Pure 4 exam papers.

本练习集来自 PDF Joiner (4) 第137页，重点复习爱德思A-Level数学大纲中得分率较高的一个主题：把有理函数分解为部分分式，然后对结果进行积分。这里涉及的技巧在 Pure 3 和 Pure 4 试卷中高频出现。

1. Overview of Partial Fractions and Integration | 部分分式与积分概述

Partial fraction decomposition is a technique that reverses the addition of rational expressions. It allows a single rational function to be written as a sum of simpler fractions whose denominators are the factors of the original denominator.

部分分式分解是一种逆转有理式加法运算的技巧。它能把一个有理函数写成若干个较简单分式的和，这些分式的分母是原分母的因式。

Once the expression is decomposed, each term can be integrated using standard results such as the natural logarithm or the inverse tangent. This is particularly useful because most rational functions cannot be integrated directly by inspection.

一旦完成分解，每一项都可以使用标准结果来积分，例如自然对数或反正切。这特别有用，因为大多数有理函数无法通过直接观察积出结果。

∫ 1/(ax+b) dx = (1/a) ln|ax+b| + C

This standard integral is the main tool after decomposition, and it should be memorised carefully.

这个标准积分是分解之后的主要工具，应当牢牢记住。

2. When to Use Partial Fraction Decomposition | 何时使用部分分式分解

Use partial fractions when the integrand is a rational function N(x)/D(x) whose denominator can be factorised into linear or irreducible quadratic factors. A simple substitution often does not work in these cases, so decomposition is the most reliable method.

当被积函数是有理函数 N(x)/D(x)，且分母可以分解为线性因式或不可约二次因式时，使用部分分式。此时简单换元往往无效，因此分解是最可靠的方法。

Before starting, always check the degree of the numerator and the degree of the denominator. The method only applies directly to proper fractions.

开始之前，务必检查分子和分母的次数。该方法仅直接适用于真分式。