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

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

Integration by substitution is a central A2 technique in Edexcel Pure Mathematics. It reverses the chain rule and allows you to integrate composite functions by changing the variable from x to a chosen u. This revision guide covers method, limit changes, trigonometric cases and examiner-style tips.

换元积分法是 Edexcel 纯数学 A2 阶段的核心技巧。它通过把变量从 x 换成选定的 u，反转链式法则，帮助你积分复合函数。本复习指南覆盖方法、定积分换限、三角换元以及考试风格的技巧。

1. When to Use Substitution | 何时使用换元法

Use substitution when an integrand contains a composite function f(g(x)) multiplied, or nearly multiplied, by the derivative g'(x). A classic sign is a bracket raised to a power, a square root of an inner expression, or an exponential with an inner linear function.

当被积函数包含复合函数 f(g(x))，并且几乎乘以内部函数的导数 g'(x) 时，就应使用换元法。典型信号是括号的幂、内层表达式的平方根，或带有内层线性函数的指数。

Example: ∫ 2x(x²+1)⁴ dx suggests u = x²+1 because 2x is exactly the derivative of x²+1.

例如：∫ 2x(x²+1)⁴ dx 提示令 u = x²+1，因为 2x 恰好是 x²+1 的导数。

∫ f(g(x)) g'(x) dx = ∫ f(u) du

This pattern is the reverse chain rule, so substitution is the natural integration tool for it.

这种模式就是反向链式法则，因此换元法是对付它的自然积分工具。

2. Choosing the Right u | 选择合适的 u

In Edexcel questions, a good first choice is usually the inner function: the expression inside brackets, under a root, in an exponent, or in a denominator. Write down u explicitly before differentiating.

在 Edexcel 考题中，通常首选内部函数：括号内、根号下、指数中或分母中的表达式。先明确写下 u，再求导。

u = x²+1 for ∫ 2x(x²+1)⁴ dx —— 对 ∫ 2x(x²+1)⁴ dx 令 u = x²+1
u = cos x for ∫ sin x cos² x dx —— 对 ∫ sin x cos² x dx 令 u = cos x
u = ln x for ∫ (ln x)² / x dx —— 对 ∫ (ln x)² / x dx 令 u = ln x

If the derivative of u is not exactly present, you may still be able to adjust by a constant factor. For example, with u = 2x+1, the derivative is 2, so you can write dx = du/2.

如果 u 的导数并不完全出现，你仍可以通过常数因子进行调整。例如，当 u = 2x+1 时，导数为 2，因此可以写成 dx = du/2。

3. The Core Substitution Steps | 换元法核心步骤

Always follow four steps: choose u, differentiate to find du/dx, replace dx by du / (du/dx), then rewrite the whole integral in terms of u only.

始终遵循四步：选择 u，求导得到 du/dx，用 du / (du/dx) 替换 dx，然后将整个积分仅用 u 重写。

u = g(x), du/dx = g'(x), dx = du / g'(x)

Then integrate with respect to u and finally substitute back if the integral is indefinite.

然后对 u 积分，若是不定积分最后再代回原变量。

Never try to mix x and u in the same integral. The substitution is only complete when every x, including dx, has been replaced by u and du.

切勿在同一个积分中混合使用 x 和 u。只有当每个 x，包括 dx，都被 u 和 du 替换后，换元才算完成。

4. Worked Indefinite Example | 不定积分示例

Find ∫ 2x(x²+1)⁴ dx. Let u = x²+1, so du/dx = 2x. The integral becomes ∫ u⁴ du.

求 ∫ 2x(x²+1)⁴ dx。令 u = x²+1，则 du/dx = 2x。积分变为 ∫ u⁴ du。

∫ u⁴ du = u⁵/5 + C = (x²+1)⁵/5 + C

Never leave the answer in terms of u for an indefinite integral; return to the original variable x and include the constant of integration.

不定积分绝不能用 u 作为最终答案；要回到原变量 x，并加上积分常数。

This example is typical of Edexcel questions where the derivative of the bracket appears exactly as a multiplier.

这个例子是 Edexcel 考题中的典型情形：括号的导数恰好作为乘数出现。

5. Definite Integrals and Changing Limits | 定积分与换限

For definite integrals, you have two options: change the limits to u-values and evaluate directly, or substitute back to x before applying the original limits. The cleaner method is to change limits immediately.

对于定积分，有两种选择：把上下限换成 u 值后直接计算，或先代回 x 再使用原上下限。较简洁的方法是立即换限。

Example: ∫ from 0 to 1 of 2x(x²+1)⁴ dx. When x=0, u=1; when x=1, u=2. Therefore:

例如：∫ 从0到1 的 2x(x²+1)⁴ dx。当 x=0 时 u=1；当 x=1 时 u=2。因此：

∫₀¹ 2x(x²+1)⁴ dx = ∫₁² u⁴ du = [u⁵/5]₁² = 32/5 – 1/5 = 31/5

Notice no constant C is needed for definite integrals.

注意定积分不需要加常数 C。

If you substitute back to x, you must use the original x-limits; if you stay in u, you must use the new u-limits. Mixing the two is a common mark-losing error.

如果代回 x，则必须使用原来的 x 上下限；如果保留 u，则必须使用新的 u 上下限。混用两者是常见的失分错误。

6. Trigonometric Substitutions | 三角换元

Trigonometric integrals often rely on identities such as sin²x + cos²x = 1, 1 + tan²x = sec²x, or double-angle forms. When a power of sine is multiplied by cosine, try u = sin x; when a power of cosine is multiplied by sine, try u = cos x.

三角积分常依赖恒等式，如 sin²x + cos²x = 1、1 + tan²x = sec²x 或倍角公式。当正弦的幂乘以余弦时，尝试 u = sin x；当余弦的幂乘以正弦时，尝试 u = cos x。

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

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