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

Integration by substitution is one of the most important techniques in the Edexcel A-Level Pure Mathematics specification. It reverses the chain rule and allows students to integrate composite functions that cannot be expanded easily. This article covers indefinite and definite integrals, choosing substitutions, trigonometric cases, and exam-ready strategies.

换元积分法是爱德思A-Level纯数学考纲中最重要的技巧之一。它逆向使用链式法则，让学生能够处理不易展开的复合函数积分。本文涵盖不定积分与定积分、如何选择换元、三角换元以及应试策略。

1. Why Use Substitution? | 为什么使用换元法

In Edexcel A-Level Pure Mathematics, integration questions often include composite functions such as (3x+2)⁵, sin(2x+1), or e^(x²). Substitution reverses the chain rule and converts these into standard integrals.

在爱德思A-Level纯数学中，积分题常包含复合函数，如 (3x+2)⁵、sin(2x+1) 或 e^(x²)。换元法逆用链式法则，将它们化为标准积分。

Expansion or trigonometric identities can work for small powers, but they are impractical for higher powers or non-polynomial forms. Substitution gives a clear, efficient route and is often specified directly in exam questions.

低次幂可以展开或使用三角恒等式，但高次幂或非多项式形式会非常繁琐。换元法提供了一条清晰高效的路径，考试中也经常直接指定换元。

∫ f'(g(x))g'(x) dx = f(g(x)) + C

2. The Reverse Chain Rule | 逆链式法则

If y = f(u) and u = g(x), then the chain rule gives dy/dx = f'(u) × u’. Integrating both sides with respect to x gives ∫ f'(u)u’ dx = f(u) + C. Therefore the inner derivative u’ is essential for the method.

若 y = f(u)，u = g(x)，则链式法则给出 dy/dx = f'(u) × u’。对 x 积分可得 ∫ f'(u)u’ dx = f(u) + C。因此内层导数 u’ 是换元法的关键。

When we set u = g(x), we also write du = u’ dx, or equivalently dx = du/u’. This converts the integration variable from x to u and restores the standard integral form.

设 u = g(x) 时，同时写出 du = u’ dx，或等价地 dx = du/u’。这样积分变量就从 x 转换为 u，恢复标准积分形式。

3. Choosing u | 如何选择 u

If the question gives a substitution, use it exactly. Otherwise choose u as the inner part of a composite, the denominator, the exponent, or the argument of a trigonometric function. Look for the derivative of u appearing in the integrand, up to a constant factor.

若题目给出换元，请严格使用。否则选择复合函数的内层、分母、指数或三角函数的内部作为 u。寻找 u 的导数是否出现在被积函数中，允许相差常数倍。

• u = inside a bracket: (ax+b)ⁿ — 括号内部
• u = denominator: 1/(ax+b) — 分母
• u = inside a root: √(g(x)) — 根号内部
• u = exponent: e^(g(x)) — 指数部分
• u = trigonometric argument: sin(g(x)) — 三角函数内部

4. Substitution in Indefinite Integrals | 不定积分中的换元

Use a clear four-step process: choose u, differentiate to find du/dx, replace all x terms and dx by u and du, integrate, then substitute back to x. If any x terms remain, express them in terms of u before integrating.

采用清晰的四步法：选择 u，求 du/dx，用 u 和 du 替换所有含 x 的项与 dx，积分，最后换回 x。如果仍有 x 项残留，先将其用 u 表示再积分。

For example, ∫ x(x²+3)⁴ dx works well because the derivative of x²+3 is 2x, and the integrand contains x, which only differs by a constant factor.

例如 ∫ x(x²+3)⁴ dx 适合换元，因为 x²+3 的导数是 2x，被积函数中含有 x，仅相差常数倍。

5. Worked Example: Indefinite Integral | 不定积分例题

Integrate ∫ x²(x³+1)⁴ dx. Let u = x³+1, then du/dx = 3x², so x

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