Integration by Substitution | 换元积分法

📚 Integration by Substitution | 换元积分法

Integration by substitution is a core technique in Edexcel A Level Mathematics, especially in Pure Mathematics 3 and 4. It turns complicated integrals into standard forms by changing the variable of integration. This article explains the method, gives worked examples, and highlights common exam pitfalls.

换元积分法是 Edexcel A Level 数学（尤其是 Pure Mathematics 3 和 4）的核心技巧。它通过改变积分变量，把复杂积分化为标准形式。本文讲解方法、给出例题，并指出常见的考试失分点。

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

Integration by substitution is the reverse process of the chain rule for differentiation. When an integrand has the form f(g(x)) × g'(x), substituting u = g(x) reduces it to ∫ f(u) du.

换元积分法是链式法则求导的逆过程。当被积函数具有 f(g(x)) × g'(x) 的形式时，令 u = g(x) 可将其化为 ∫ f(u) du。

∫ f(g(x)) × g'(x) dx = ∫ f(u) du, where u = g(x)

This technique is essential for A Level exam questions involving powers of linear functions, trigonometric products, exponential functions, and rational expressions. You will see it in both indefinite and definite integrals.

这一技巧对 A Level 考试中涉及线性函数幂、三角乘积、指数函数和有理式的题目非常重要。在不定积分和定积分中都会遇到。

To apply the method accurately, you must rewrite the entire integral in terms of u, including dx and any limits. Careful working is needed to earn full method marks in Edexcel papers.

要准确使用该方法，必须把整个积分都用 u 重写，包括 dx 和所有上下限。在 Edexcel 试卷中，细致的解题步骤是获得满分方法分的关键。

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

The chain rule states that d/dx [F(g(x))] = F'(g(x)) × g'(x). Therefore, if you recognise F'(g(x)) × g'(x) in an integral, the answer is simply F(g(x)) + C.

链式法则指出 d/dx [F(g(x))] = F'(g(x)) × g'(x)。因此，如果能在积分中识别出 F'(g(x)) × g'(x)，答案就是 F(g(x)) + C。

For example, since d/dx (x² + 1)⁵ = 5(x² + 1)⁴ × 2x = 10x(x² + 1)⁴, reversing this gives ∫ 10x(x² + 1)⁴ dx = (x² + 1)⁵ + C.

例如，因为 d/dx (x² + 1)⁵ = 5(x² + 1)⁴ × 2x = 10x(x² + 1)⁴，反过来可得 ∫ 10x(x² + 1)⁴ dx = (x² + 1)⁵ + C。

∫ 10x(x² + 1)⁴ dx = (x² + 1)⁵ + C

When the constant multiple is not exact, you can adjust by a scalar. For example, ∫ x(x² + 1)⁴ dx = (1/10)(x² + 1)⁵ + C. This recognition speed is useful, but formal substitution is safer in exams.

当常数倍数不完全一致时，可以通过标量进行调整。例如，∫ x(x² + 1)⁴ dx = (1/10)(x² + 1)⁵ + C。这种快速识别很有用，但在考试中使用规范代换更稳妥。

3. Choosing the Substitution u | 如何选择代换 u

A good substitution is usually the inner function of a composite expression, or a function whose derivative appears elsewhere in the integrand. Look for a term inside brackets, under a root, in a denominator, or as an exponent.

一个好的代换通常是复合表达式的内层函数，或者其导数出现在被积函数其他位置的函数。寻找括号内、根号下、分母中或指数上的项。

If you see (ax + b)ⁿ, try u = ax + b.
If you see √(f(x)), try u = f(x).
If you see e^{f(x)}, try u = f(x).
If you see a product of a function and its derivative, try the function as u.

中文列表：

若看到 (ax + b)ⁿ，可尝试 u = ax + b。
若看到 √(f(x))，可尝试 u = f(x)。
若看到 e^{f(x)}，可尝试 u = f(x)。
若看到一个函数与其导数的乘积，可尝试令该函数为 u。

In Edexcel exams, the substitution is often given in the question for harder integrals. When it is not given, the structure of the integrand must guide your choice.

在 Edexcel 考试中，较难的积分通常会给出代换。如果未给出，必须根据被积函数的结构来选择合适的代换。

A quick check is to compute du/dx. If the remaining part of the integrand is not close to du/dx, the chosen substitution may not be efficient.

一个快速检验方法是计算 du/dx。如果被积函数剩余部分与 du/dx 相差较大，所选的代换可能并不高效。

4. Worked Example: Polynomial Substitution | 多项式代换例题

Worked example: Find ∫ 6x(x² + 3)⁷ dx.

例题：求 ∫ 6x(x² + 3)⁷ dx。

Let u = x² + 3. Then du/dx = 2x, so du = 2x dx. The integral becomes ∫ 6x × u⁷ × (du / 2x) = ∫ 3u⁷ du.

令 u = x² + 3，则 du/dx = 2x，所以 du = 2x dx。原积分变为 ∫ 6x × u⁷ × (du / 2x) = ∫ 3u⁷ du。

∫ 3u⁷ du = 3u⁸ / 8 + C

Substitute back u = x² + 3 to obtain the final answer in terms of x.

将 u = x² + 3 代回，得到以 x 表示的最终答案。

∫ 6x(x² + 3)⁷ dx = 3(x² + 3)⁸ / 8 + C

This type of question is common on Edexcel P3 and P4 papers, often worth four to six marks. Marks are usually awarded for correct substitution, correct integration in u, and correct back substitution.

这类题目常见于 Edexcel P3 和 P4 试卷，通常占 4 到 6 分。评分通常分配给正确代换、对 u 正确积分，以及正确代回原变量。

5. Trigonometric Substitution Cases | 三角代换情形

Trigonometric integrals often require substitution when one trig function is the derivative of another, or when powers of sin x and cos x appear together.

当一个三角函数是另一个三角函数的导数，或者 sin x 和 cos x 的幂同时出现时，三角积分通常需要代换。

Example: ∫ sin³x cos x dx. Let u = sin x, so du/dx = cos x, hence du = cos x dx. The integral becomes ∫ u³ du = u⁴ / 4 + C = sin⁴x / 4 + C.

例：∫ sin³x cos x dx。令 u = sin x，则 du/dx = cos x，所以 du = cos x dx。积分变为 ∫ u³ du = u⁴ / 4 + C = sin⁴x / 4 + C。

∫ sin³x cos x dx = sin⁴x / 4 + C

For even powers, such as ∫ sin²x dx or ∫ cos²x dx, use double-angle identities instead of substitution. Edexcel students must distinguish between these cases.

对于偶次幂，如 ∫ sin²x dx 或 ∫ cos²x dx，应使用倍角恒等式而不是代换。Edexcel 考生必须区分这些情形。

Use substitution for odd powers with one spare derivative.
Use double-angle formulas for even powers.

中文列表：

奇次幂且有余下的导数时用代换。
偶次幂用倍角公式。

Also, for integrals such as ∫ tan x dx, writing tan x = sin x / cos x and using u = cos x leads to -ln |cos x| + C.

此外，对于 ∫ tan x dx，写出 tan x = sin x / cos x 并令 u = cos x，可得到 -ln |cos x| + C。

6. Definite Integrals and Changing Limits | 定积分与上下限变换

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

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Integration by Substitution | 换元积分法