Partial Fractions

Core Sentence / 核心句

Partial fractions decomposes a proper rational function into simpler rational pieces with elementary antiderivatives.

中文理解：部分分式把一个 proper rational function 拆成若干简单分式，使每一项都能直接得到 logarithmic 或 inverse-trigonometric antiderivative。

Basic Pattern / 基本结构

For distinct linear factors,

$$\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}.$$

Workflow / 流程

check degree
-> long divide if improper
-> factor denominator
-> write full decomposition template
-> solve coefficients
-> recombine to verify
-> integrate each piece

Watch Out / 易错点

• Partial fractions requires a proper fraction first.
• Repeated linear factors need every power from $1$ through the multiplicity.
• Irreducible quadratic factors require linear numerators.
• Absolute values remain in logarithmic antiderivatives.
• Always verify coefficients by recombining the decomposition.

Source / 出现位置

• 第9课 · 使用条件
• 第9课 · 系数求解
• 第9课 · 平方差分解

Connected Notes / 连接

• Integrals
• Antiderivative
• Integration by Parts
• 微积分BC第9次正课-积分方法与反常积分