Integration by Substitution | 换元积分法

📚 Integration by Substitution | 换元积分法

Integration by substitution is one of the most powerful techniques in Edexcel A-Level Pure Mathematics. It allows you to reverse the chain rule and integrate complicated expressions by changing the variable to make the integral much simpler.

换元积分法是 Edexcel A-Level 纯数学中最强大的技巧之一。它可以让你反过来使用链式法则，通过改变变量使被积函数更简单，从而对复杂表达式进行积分。

1. What is Integration by Substitution? | 什么是换元积分法？

In integration by substitution, you replace a difficult expression in x with a new variable u. The derivative du/dx is then used to rewrite dx in terms of du. This transforms the original integral into one in u that is easier to evaluate.

在换元积分法中，你把 x 中较难处理的部分替换成一个新变量 u。然后利用导数 du/dx 把 dx 改写成 du 的形式。这样原积分就转化为一个关于 u 的更容易计算的积分。

You have already used substitution when you integrated functions such as (ax + b)ⁿ by thinking of the inside function as u. At A-Level, the method is extended to more general expressions, including products, quotients, exponentials and trigonometric combinations.

你在积分 (ax + b)ⁿ 这类函数时已经用过代换思想，把内层函数看作 u。在 A-Level 阶段，这一方法会推广到更一般的表达式，包括乘积、商、指数函数和三角函数组合。

2. The Reverse Chain Rule | 反向链式法则

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