Integration by Parts

Core Sentence / 核心句

Integration by parts reverses the Product Rule to replace one product integral with a simpler one.

中文理解：分部积分把乘积求导公式倒过来使用，让一个难处理的乘积积分转化成更简单的积分。

Formula / 公式

$$\int udv=uv-\int vdu.$$

It follows from

$$d(uv)=udv+vdu.$$

Selection Rule / 选择规则

Choose $u$ so differentiation simplifies it; choose $dv$ so integration remains manageable. LIATE is a heuristic, not a theorem.

Workflow / 流程

identify product
-> choose u and dv
-> compute du and v
-> write uv - integral(v du)
-> confirm the new integral is simpler
-> integrate and differentiate to verify

Watch Out / 易错点

• $dv$ includes the differential and must be integrated to obtain $v$.
• Do not choose $u$ so that $du$ becomes more complicated.
• Keep the minus sign before $\int vdu$.
• Definite integrals require bounds on both the $uv$ term and remaining integral.
• A product alone does not guarantee that by parts is the best method.

Source / 出现位置

• 第9课 · Product Rule 逆运算
• 第9课 · x ln x
• 第9课 · abstract by parts

Connected Notes / 连接

• Antiderivative
• Integrals
• U-Substitution
• 微积分BC第9次正课-积分方法与反常积分