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 for evaluating integrals. It reverses the chain rule and transforms a complicated integral into a standard form that can be integrated directly. In Edexcel A-Level Mathematics, substitution questions often involve powers of linear or quadratic functions, products with a derivative present, and definite integrals where the limits must be changed.

换元积分法是求解积分最有力的技巧之一。它反转链式法则，将复杂积分转化为可以直接积分的标准形式。在 Edexcel A-Level 数学中，换元题常涉及线性或二次函数的幂、被积函数中含有导数乘积，以及需要更换上下限的定积分。

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

Integration by substitution is a method for finding indefinite and definite integrals by introducing a new variable u to simplify the integrand. The idea is to choose u = g(x) so that the differential du = g'(x) dx replaces part of the original integral.

换元积分法是通过引入新变量 u 来简化被积函数，从而求解不定积分和定积分的方法。核心思想是选取 u = g(x)，使得微分 du = g'(x) dx 替换原积分中的一部分。

Formally, if u = g(x), then du/dx = g'(x), so du = g'(x) dx. The integral ∫ f(g(x)) g'(x) dx becomes ∫ f(u) du after substitution. This turns an unfamiliar composite expression into a basic integral with respect to u.

形式上，如果 u = g(x)，则 du/dx = g'(x)，因此 du = g'(x) dx。换元后，积分 ∫ f(g(x)) g'(x) dx 变为 ∫ f(u) du。这样就将不熟悉的复合表达式转化为关于 u 的基本积分。

A simple example is ∫ 2x(x² + 1)⁵ dx. Let u = x² + 1, then du = 2x dx. The integral becomes ∫ u⁵ du, which is easy to evaluate.

一个简单例子是 ∫ 2x(x² + 1)⁵ dx。令 u = x² + 1，则 du = 2x dx。积分变为 ∫ u⁵ du，容易计算。

2. The Reverse Chain Rule Idea | 反向链式法则思想

Differentiation uses the chain rule: if y = f(u) and u = g(x), then dy/dx = f'(u) g'(x). Integration by substitution works backwards: if the integrand is of the form f'(g(x)) g'(x), its antiderivative is f(g(x)) + C.

微分运用链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = f'(u) g'(x)。换元积分法反向操作：如果被积函数形如 f'(g(x)) g'(x)，则其原函数为 f(g(x)) + C。

For example, since d/dx (x² + 1)⁶ = 6(x² + 1)⁵ × 2x = 12x(x² + 1)⁵, it follows that ∫ 12x(x² + 1)⁵ dx = (x² + 1)⁶ + C. The substitution u = x² + 1 makes this connection explicit.

例如，因为 d/dx (x² + 1)⁶ = 6(x² + 1)⁵ × 2x = 12x(x² + 1)⁵，所以 ∫ 12x(x² + 1)⁵ dx = (x² + 1)⁶ + C。使用 u = x² + 1 的代换可以明确展示这一联系。

This reverse chain rule view is very useful in exams because many Edexcel questions ask you to spot an inner function and its derivative multiple within the integrand.

这种反向链式法则的观点在考试中非常有用，因为许多 Edexcel 题目要求你识别被积函数中的内层函数及其导数的倍数。

3. Choosing the Substitution u | 选择换元变量 u

The key skill in substitution is identifying the inner function of a composite expression. Typically u is chosen as the function inside brackets, under a root, in the denominator, or as the exponent.

换元的核心技能是识别复合表达式的内层函数。通常选择括号内的函数、根号下的函数、分母中的函数或指数部分作为 u。

Common choices include: u = ax + b for powers of linear functions; u = x² + 1 for integrals like ∫ x(x² + 1)⁵ dx; u = cos x for integrals like ∫ sin x cos²x dx.

常见选择包括：线性函数幂用 u = ax + b；对 ∫ x(x² + 1)⁵ dx 取 u = x² + 1；对 ∫ sin x cos²x dx 取 u = cos x。

Integrand form	Suggested u
(ax + b)ⁿ, with n ≠ -1	u = ax + b
f'(x) / f(x)	u = f(x)
sin³x cos x	u = sin x
x / (x² + 1)	u = x² + 1

被积函数形式与推荐代换：括号线性函数的幂用 u = ax + b；f'(x)/f(x) 用 u = f(x)；sin³x cos x 用 u = sin x；x/(x²+1) 用 u = x²+1。

4. The Standard Substitution Process | 标准换元步骤

For an indefinite integral, follow four steps: choose u, differentiate to find du/dx, replace dx by du / (du/dx), then integrate with respect to u and finally substitute back for x.

对于不定积分，按照四步进行：选取 u，对 u 求导得到 du/dx，将 dx 替换为 du / (du/dx)，然后对 u 积分，最后将 u 代回 x。

∫ f(g(x)) g'(x) dx = ∫ f(u) du

In practice, you should write du = g'(x) dx explicitly. If the original integrand is missing a constant multiple of g'(x), you may need to adjust by a numerical factor.

实际操作中，应明确写出 du = g'(x) dx。如果原被积函数缺少 g'(x) 的常数倍数，可能需要通过数值因子调整。

Step 1: Let u = g(x).
Step 2: Find du/dx = g'(x), so du = g'(x) dx.
Step 3: Substitute all x expressions and dx in terms of u and du.
Step 4: Integrate and replace u with g(x).

中文顺序：第1步：设 u = g(x)。第2步：求 du/dx = g'(x)，故 du = g'(x) dx。第3步：将所有 x 表达式和 dx 用 u 和 du 替换。第4步：对 u 积分并将 u 代回 g(x)。

5. Definite Integrals and Changing Limits | 定积分与更换上下限

When evaluating a definite integral by substitution, you must change the limits of integration from x-values to u-values. This avoids having to substitute back before applying limits.

用换元法计算定积分时，必须将积分上下限从 x 值转换为 u 值。这样可避免在代入上下限之前再替换回 x。

If x = a gives u = g(a) and x = b gives u = g(b), then ∫ₐᵇ f(g(x)) g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du.

如果 x = a 对应 u = g(a)，x = b 对应 u = g(b)，则 ∫ₐᵇ f(g(x)) g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du。

Never mix x-limits with

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Integration by Substitution | 换元积分法