A-Level Maths: Integration by Parts Techniques | A-Level 数学：分部积分法解题技巧

📚 A-Level Maths: Integration by Parts Techniques | A-Level 数学：分部积分法解题技巧

Integration by parts is one of the most powerful techniques in A-Level calculus. It allows us to integrate products of functions by reversing the product rule for differentiation. This article explains the method step by step, with worked examples and exam-style tips.

分部积分法是 A-Level 微积分中最强大的技巧之一。它通过逆转乘积法则，帮助我们积分两个函数的乘积。本文将逐步讲解这一方法，并附有典型例题和应试技巧。

1. What is Integration by Parts? | 1. 什么是分部积分法？

Integration by parts is a technique for integrating products of functions. It is the reverse of the product rule for differentiation and is particularly useful when one factor can be differentiated repeatedly and the other can be integrated easily.

分部积分法是处理两个函数乘积的积分技巧。它是微分乘积法则的逆运算，特别适用于一个因子可以反复求导、另一个因子容易积分的情形。

For example, integrals such as ∫ x eˣ dx, ∫ x cos x dx and ∫ ln x dx all require this method in A-Level Maths.

例如，∫ x eˣ dx、∫ x cos x dx 和 ∫ ln x dx 这类积分在 A-Level 数学中都需要用到分部积分法。

2. The Formula and Its Derivation | 2. 公式及其推导

The formula for integration by parts is:

分部积分法的公式为：

∫ u dv = u v − ∫ v du

In this formula, we choose u and dv from the integrand. The symbol dv represents a differential, such as eˣ dx or cos x dx.

在公式中，我们从被积函数中选取 u 和 dv。符号 dv 表示一个微分，例如 eˣ dx 或 cos x dx。

This formula comes from the product rule for differentiation:

该公式来源于微分中的乘积法则：

d/dx (u v) = u dv/dx + v du/dx

Integrating both sides with respect to x gives:

两边对 x 积分得到：

u v = ∫ u dv + ∫ v du

Rearranging this result gives the integration by parts formula.

整理后即可得到分部积分公式。

3. Choosing u and dv: The LIATE Rule | 3. 选择 u 和 dv：LIATE 规则

The most common question is: which

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