📚 Reverse Chain Rule for Integration | 反向链式法则求积分
The reverse chain rule is a fundamental integration technique in A-Level Mathematics. It allows us to integrate composite functions by reversing the chain rule used in differentiation, saving us from expanding brackets or performing lengthy substitutions.
反向链式法则是A-Level数学中一项基础积分技巧。它通过逆转微分中的链式法则来对复合函数积分，从而避免展开括号或进行冗长的换元。
This article explains the reverse chain rule from first principles, shows how to recognise its key patterns, and provides worked examples across polynomial, exponential, trigonometric and rational functions. By the end, you will be able to apply this powerful method confidently in both pure and applied contexts.
本文将从头讲解反向链式法则，展示如何识别其关键模式，并给出多项式、指数、三角函数和有理函数的例题。学完后，你将能够在纯数学和应用情境中自信地运用这一强大方法。
1. What is the Reverse Chain Rule? | 什么是反向链式法则
In differentiation, the chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). The reverse chain rule works in the opposite direction: when an integrand contains an expression of the form f'(g(x)) · g'(x), we can immediately recognise its antiderivative as f(g(x)).
微分中，链式法则指出：若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。反向链式法则则反向运作：当被积函数含有 f'(g(x)) · g'(x) 的形式时，我们可立刻识别其原函数为 f(g(x))。
For instance, the derivative of (x² + 1)³ is 3(x² + 1)² · 2x = 6x(x² + 1)². Therefore, integrating 6x(x² + 1)² must give back (x² + 1)³ plus a constant.
例如，(x² + 1)³ 的导数为 3(x² + 1)² · 2x = 6x(x² + 1)²。因此，对 6x(x² + 1)² 积分必定还原为 (x² + 1)³ 加上常数。
2. The Core Pattern: ∫ f'(x)[f(x)]ⁿ dx | 核心模式：∫ f'(x)[f(x)]ⁿ dx
Consider the derivative of [f(x)]ⁿ⁺¹. Using the chain rule:
考虑 [f(x)]ⁿ⁺¹ 的导数。运用链式法则：
d/dx [f(x)]ⁿ⁺¹ = (n + 1) · f'(x) · [f(x)]ⁿ
Dividing both sides by (n + 1) gives the general reverse chain rule formula for powers:
两边同时除以 (n + 1)，得到幂函数的反向链式法则一般公式：
∫ f'(x) [f(x)]ⁿ dx = [f(x)]ⁿ⁺¹ / (n + 1) + c, n ≠ −1
This formula applies whenever the integrand contains a function f(x) and its derivative f'(x) as a factor, possibly up to a constant multiple.
只要被积函数中含有函数 f(x) 及其导数 f'(x) 作为因子（相差常数倍数也可），该公式即可适用。
3. Recognising the Derivative Factor | 识别导数因子
The most important skill is spotting f'(x) inside the integrand. Start by choosing a convenient inner function f(x), then check whether its derivative appears outside the brackets.
最关键的技巧是被积函数中识别出 f'(x)。先选取一个合适的内层函数 f(x)，然后检查其导数是否出现在括号外。
-
Look for a function inside a bracket or inside a function (such as sin, cos, ln, e), then test its derivative.
寻找括号内或嵌套在函数（如 sin、cos、ln、e）内部的函数，然后检验其导数。
-
If the derivative appears exactly, apply the formula directly.
若导数精确出现，直接套用公式。
-
If the derivative appears multiplied by a constant k, factor out 1/k before integrating.
若导数出现时带有一个常数倍数 k，则在积分前提出因子 1/k。
Example: ∫ 2x cos(x²) dx. Here f(x) = x² and f'(x) = 2x, so the integral is sin(x²) + c.
例：∫ 2x cos(x²) dx。此处 f(x) = x²，f'(x) = 2x，因此积分为 sin(x²) + c。
4. Linear Functions: A Special Case | 线性函数特例
When the inner function is linear, f(x) = ax + b, then f'(x) = a is a constant. The reverse chain rule simplifies to:
当内层函数为线性函数 f(x) = ax + b 时，f'(x) = a 为常数。反向链式法则简化为：
∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n + 1)] + c
For example, ∫ (3x − 1)⁴ dx = (3x − 1)⁵ / (3 × 5) + c = (3x − 1)⁵ / 15 + c.
例如，∫ (3x − 1)⁴ dx = (3x − 1)⁵ / (3 × 5) + c = (3x − 1)⁵ / 15 + c。
The same idea extends to trigonometric and exponential functions with linear arguments:
同样思想可推广至线性参数的三角函数和指数函数：
| ∫ sin(ax + b) dx = −(1/a) cos(ax + b) + c |
| ∫ cos(ax + b) dx = (1/a) sin(ax + b) + c |
| ∫ e^(ax + b) dx = (1/a) e^(ax + b) + c |
These formulas are essential for quick integration in exam questions.
这些公式在考试题目中可快速完成积分，极为重要。
5. Exponential Functions: ∫ f'(x) e^(f(x)) dx | 指数函数：∫ f'(x) e^(f(x)) dx
For an exponential function e^(f(x)), the derivative is f'(x) e^(f(x)). Reversing this gives:
对于指数函数 e^(f(x))，其导数为 f'(x) e^(f(x))。逆转后得到：
∫ f'(x) e^(f(x)) dx = e^(f(x)) + c
This result holds for any differentiable function f(x), not only linear ones.
该结论适用于任何可微函数 f(x)，不仅限于线性函数。
Example: ∫ x e^(x²) dx. Let f(x) = x² so f'(x) = 2x. Since the integrand contains x rather than 2x, rewrite the integral as (1/2) ∫ 2x e^(x²) dx = (1/2) e^(x²) + c.
例：∫ x e^(x²) dx。令 f(x) = x²，则 f'(x) = 2x。由于被积函数含 x 而非 2x，将积分改写为 (1/2) ∫ 2x e^(x²) dx = (1/2) e^(x²) + c。
Always check the coefficient: if the derivative factor is missing a constant, add the reciprocal outside the integral.
务必检查系数：若导数因子缺少常数倍，在积分号外补上其倒数。
6. Trigonometric Functions | 三角函数
Reverse chain rule is especially useful for trigonometric composite functions. The standard results are:
对三角复合函数，反向链式法则尤其有用。标准结果如下：
-
∫ f'(x) sin(f(x)) dx = −cos(f(x)) + c
该式对应 sin(f(x)) 的复合积分。
-
∫ f'(x) cos(f(x)) dx = sin(f(x)) + c
该式对应 cos(f(x)) 的复合积分。
-
∫ f'(x) sec²(f(x)) dx = tan(f(x)) + c
Published by TutorHao | Mathematics Revision Series | aleveler.com更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导