Reverse Chain Rule for Integration | 积分中的反链式法则

📚 Reverse Chain Rule for Integration | 积分中的反链式法则

In Edexcel A-Level Mathematics, the reverse chain rule is a core integration technique used when the integrand is, or can be adjusted to become, the derivative of a composite function multiplied by a constant. It is essentially the chain rule for differentiation applied in reverse, and it allows many standard integrals to be evaluated quickly without a full substitution.

在 Edexcel A-Level 数学中，反链式法则是一种核心积分方法。当被积函数是某个复合函数导数的常数倍，或者可以调整成这种形式时，就可以使用反链式法则。它本质上是微分链式法则的逆运算，能够让学生无需完整换元就能快速求出许多标准积分。

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1. What Is the Reverse Chain Rule? | 什么是反链式法则？

The chain rule for differentiation states that if F and f are differentiable functions, then d/dx [F(f(x))] = f'(x) F'(f(x)). Integrating both sides of this identity gives the reverse chain rule: if we can spot the pattern f'(x) F'(f(x)) inside an integral, then the answer is simply F(f(x)) + C.

微分中的链式法则表明，如果 F 和 f 都是可导函数，那么 d/dx [F(f(x))] = f'(x) F'(f(x))。对这个恒等式两边同时积分，就得到反链式法则：如果我们能在被积函数中发现 f'(x) F'(f(x)) 这样的模式，那么积分结果就是 F(f(x)) + C。

d/dx [F(f(x))] = f'(x) F'(f(x)) ⟹ ∫ f'(x) F'(f(x)) dx = F(f(x)) + C

This means the reverse chain rule is not a separate rule from the chain rule, but the chain rule read backwards. In many exam questions, the fastest method is to identify the inner function f(x), its derivative f'(x), and then apply the appropriate standard form.

这意味着反链式法则并不是与链式法则完全独立的另一条法则，而是把链式法则反过来读。在许多考试题目中，最快的方法就是先确定内层函数 f(x) 和它的导数 f'(x)，然后套用相应的标准积分公式。

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2. The Core Formula and Conditions | 核心公式与使用条件

The most common power form of the reverse chain rule is used when the integrand contains a power of a function f(x) multiplied by its derivative f'(x). If k is a constant and n ≠ −1, then the integral is evaluated by increasing the power by 1 and dividing by the new power.

反链式法则最常见的幂函数形式适用于被积函数含有函数 f(x) 的幂，并且乘上它的导数 f'(x) 的情形。如果 k 是常数且 n ≠ −1，那么积分可以通过把指数加 1，再除以新的指数来求得。

∫ k f'(x) [f(x)]ⁿ dx = k [f(x)]ⁿ⁺¹/(n+1) + C, n ≠ −1

For this formula to be used directly, the derivative of the inner function f(x) must be present in the integrand exactly, or be present up to a constant multiple. If the derivative is missing but can be introduced by multiplying and dividing by a constant, the same rule still works.

要直接使用这个公式，内层函数 f(x) 的导数必须精确地出现在被积函数中，或者至多相差一个常数倍。如果导数没有直接出现，但可以通过乘以并除以一个常数来构造出来，那么该法则仍然适用。

The table below summarises the five standard patterns most commonly tested in the Edexcel A-Level specification.

下表总结了 Edexcel A-Level 考纲中最常见的五种标准反链式法则模式。

Integral form | 积分形式	Result | 结果
∫ k f'(x) [f(x)]ⁿ dx, n ≠ −1	k [f(x)]ⁿ⁺¹/(n+1) + C
∫ k f'(x) e^(f(x)) dx	k e^(f(x)) + C
∫ k f'(x) sin(f(x)) dx	−k cos(f(x)) + C
∫ k f'(x) cos(f(x)) dx	k sin(f(x)) + C
∫ k f'(x)/f(x) dx	k ln|f(x)| + C

It is important to check the value of n before using the power rule. When n = −1, the power form does not apply because division by zero would occur. Instead, the logarithmic form must be used.

在使用幂函数公式之前，一定要检查 n 的值。当 n = −1 时，幂函数公式不适用，因为会出现除以零的情况。此时必须改用对数形式的公式。

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3. Power Form: ∫ k f'(x) [f(x)]ⁿ dx | 幂函数形式

To integrate a power form, first choose the inner function u = f(x). Then differentiate it to find f'(x). Compare this derivative with the coefficient outside the bracket. If the outside coefficient is a constant multiple of f'(x), write the integral in the standard form and apply the formula.

要积分幂函数形式，首先要选择内层函数 u = f(x)。然后对它求导得到 f'(x)。将导数与括号外的系数进行比较。如果外部系数是 f'(x) 的常数倍，就把积分写成标准形式并套用公式。

For example, evaluate ∫ 2x (x² + 1)³ dx. Here f(x) = x² + 1, so f'(x) = 2x. The integrand is exactly f'(x) [f(x)]³, so n = 3 and k = 1.

例如，求 ∫ 2x (x² + 1)³ dx。这里 f(x) = x² + 1，因此 f'(x) = 2x。被积函数恰好是 f'(x) [f(x)]³，所以 n = 3，k = 1。

∫ 2x (x² + 1)³ dx = (x² + 1)⁴/4 + C

If the derivative is not exact, adjust with a constant. For ∫ x (x² + 5)⁴ dx, the inner function is f(x) = x² + 5 with f'(x) = 2x. The integrand has x, which is half of 2x, so multiply and divide by 2.

如果导数不完全匹配，可以用常数来调整。对于 ∫ x (x² + 5)⁴ dx，内层函数为 f(x) = x² + 5，其导数 f'(x) = 2x。被积函数中只有 x，它是 2x 的一半，因此乘以并除以 2。

∫ x (x² + 5)⁴ dx = ½ ∫ 2x (x² + 5)⁴ dx = ½ × (x² + 5)⁵/5 + C = (x² + 5)⁵/10 + C

For a linear inner function such as ax + b, the derivative is simply a, so the rule becomes ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n+1)] + C. This is a very common special case in exam questions.

对于线性内层函数 ax + b，其导数就是常数 a，因此公式变为 ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n+1)] + C。这是考试题目中非常常见的一种特殊情况。

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4. Exponential Form: ∫ k f'(x) e^(f(x)) dx | 指数函数形式

The reverse chain rule also applies immediately to exponential functions. If the integrand contains e^(f(x)) multiplied by f'(x), then the integral is e^(f(x)) + C, since the derivative of e^(f(x)) is f'(x) e^(f(x)).

反链式法则同样可以直接用于指数函数。如果被积函数含有 e^(f(x))，并且乘以 f'(x)，那么积分结果就是 e^(f(x)) + C，因为 e^(f(x)) 的导数正是 f'(x) e^(f(x))。

∫ k f'(x) e^(f(x)) dx = k e^(f(x)) + C

For a linear inner function f(x) = ax + b, the derivative is a, so ∫ e^(ax + b) dx = (1/a) e^(ax + b) + C. This is worth memorising because it appears frequently in pure mathematics and applied contexts.

对于线性内层函数 f(x) = ax + b，导数为 a，因此 ∫ e^(ax + b) dx = (1/a) e^(ax + b) + C。这个结论值得牢记，因为它在纯数学和应用题中经常出现。

For example, ∫ e^(5x) dx = ⅕ e^(5x) + C, because the derivative of 5x is 5, so we divide by 5. Similarly, ∫ 3 e^(3x) dx = e^(3x) + C, since the derivative 3 is already present exactly.

例如，∫ e^(5x) dx = ⅕ e^(5x) + C，因为 5x 的导数是 5，所以要除以 5。类似地，∫ 3 e^(3x) dx = e^(3x) + C，因为导数 3 已经精确出现。

If the inner function is not linear, the same principle applies. For ∫ 2x e^(x²) dx, set f(x) = x² so f'(x)

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