Integration by Substitution for Edexcel A-Level Maths | Edexcel A-Level数学换元积分法精讲

📚 Integration by Substitution for Edexcel A-Level Maths | Edexcel A-Level数学换元积分法精讲

In Edexcel A-Level Mathematics, integration by substitution is one of the most powerful techniques for evaluating integrals that cannot be tackled by standard formulae alone. This method reverses the chain rule and is examined regularly in Pure Mathematics, especially when the integrand contains a composite function multiplied by some derivative of the inner function.

在Edexcel A-Level数学中，换元积分法是处理无法仅靠标准公式求解的积分的最强大技巧之一。该方法反向运用链式法则，在纯数学考试中经常出现，尤其是当被积函数包含复合函数乘以内层函数的某阶导数时。

1. The Core Idea | 核心思想

Integration by substitution is essentially the chain rule in reverse. If you know that the derivative of f(g(x)) is f'(g(x))g'(x), then integrating f'(g(x))g'(x) with respect to x recovers f(g(x)) plus a constant. Substitution lets you rename the inner function g(x) as u so that the integral becomes much simpler.

换元积分本质上是链式法则的逆运算。如果你知道 f(g(x)) 的导数为 f'(g(x))g'(x)，那么对 f'(g(x))g'(x) 关于 x 积分就会还原为 f(g(x)) 加常数。换元可以让你把内层函数 g(x) 重新命名为 u，使积分形式变得简单得多。

On the Edexcel paper, you will often see a clue that substitution is needed: the integrand contains an ‘inner function’ and its derivative appears next to it, but the derivative may be missing a constant factor. For example, in ∫ x(x² + 5)⁶ dx, the derivative of x² + 5 is 2x, and we already have x in the integrand, so a substitution u = x² + 5 makes sense.

在Edexcel试卷中，你通常会看到需要换元积分的线索：被积函数包含一个“内层函数”，且其导数出现在旁边，只是导数可能相差一个常数因子。例如，在 ∫ x(x² + 5)⁶ dx 中，x² + 5 的导数是 2x，而被积函数中已经有 x，因此设 u = x² + 5 是合理的。

2. Choosing the Substitution | 选择替换变量

The substitution u = g(x) is usually chosen as the inner part of a composite function, the expression inside a bracket, under a root, in a denominator, or inside a trigonometric or exponential function. You should also write du/dx = g'(x) and then express dx in terms of du.

通常选择 u = g(x) 作为复合函数的内层部分，例如括号内、根号内、分母中或三角函数与指数函数内部的表达式。你还需要写出 du/dx = g'(x)，然后将 dx 用 du 表示。

For example, if u = 3x² + 1, then du/dx = 6x, so du = 6x dx. If the original integral contains x dx rather than 6x dx, you can multiply or divide by a constant to match the given integral. This step is essential before you replace every x and dx in the original problem.

例如，若 u = 3x

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）