IB Maths: Differentiating Integrals with Respect to a Parameter | IB数学：含参积分对参数求导

📚 IB Maths: Differentiating Integrals with Respect to a Parameter | IB数学：含参积分对参数求导

Most IB students meet the definite integral as a number, or as a function of its upper limit. There is a third, far more powerful viewpoint: treat some constant inside the integrand as a dial you are allowed to turn, and the integral becomes a function of that dial. Turning the dial, then asking how the value changes, is called differentiation with respect to a parameter. It converts hard integrals into easy derivatives, and it is one of the most elegant tools in the IB higher-level calculus toolkit.

大多数 IB 学生把定积分看成一个数，或者看成上限的函数。其实还有第三种、也强大得多的视角：把被积函数里某个“常数”当成一个可以转动的旋钮，积分值就成了这个旋钮的函数。转动旋钮、再问积分值如何变化，就叫“对参数求导”。它能把很难的积分转化为很容易的求导，是 IB 高等级微积分工具箱中最优雅的技巧之一。

1. What Is a Parameter Integral? | 什么是含参积分

A parameter integral is a definite integral in which a symbol a appears in the integrand, but a is treated as a constant during the integration with respect to x. The result is a function of a, usually written F(a) = ∫ f(x, a) dx over some interval. Every IB student has already met one: F(a) = ∫₀¹ xᵃ dx = 1/(a + 1) for a > −1. Here a is fixed while x runs from 0 to 1, and afterwards F becomes a function of a.

含参积分指的是：被积函数里含有符号 a，但在对 x 积分时把 a 当作常数。积分完成后结果是一个关于 a 的函数，通常写作 F(a) = ∫ f(x, a) dx（在某个区间上）。每个 IB 学生其实都见过一个：F(a) = ∫₀¹ xᵃ dx = 1/(a + 1)（a > −1）。这里积分时 a 固定，x 从 0 跑到 1，最后 F 成为 a 的函数。

The key mental switch is this: the parameter is a constant while you integrate, and a variable the moment you stop. That single sentence is the whole idea. Everything else in this article is bookkeeping about how to differentiate with respect to that dial correctly.

关键的思维切换是：积分时参数是常数，积分一结束它就是变量。这一句话就是全部思想。本文剩下的部分，都是关于“如何正确地对这个旋钮求导”的技术细节。

2. The Master Formula: Leibniz’s Rule | 主公式：莱布尼茨法则

The central result is usually called the Leibniz integral rule, or “differentiating under the integral sign”. For the simplest case, where the limits of integration are constants and the parameter appears only inside the integrand, the rule says: you may push the derivative d/da through the integral sign, provided you differentiate the integrand partially with respect to a while holding x fixed.

核心结果通常称为莱布尼茨积分法则，或“积分号下求导”。在最简单的情形中——积分上下限是常数、参数只出现在被积函数内部——法则说：你可以把导数 d/da 推进积分号里面，只要把被积函数对 a 求偏导（求导时把 x 视为常数）。

If F(a) = ∫ₐ₁ᵃ₂ f(x, a) dx with a₁, a₂ constant, then F′(a) = ∫ₐ₁ᵃ₂ (∂f/∂a) dx

The full version, which also handles limits that depend on a, adds two boundary terms and is the version you should memorise, because IB Paper 3 and olympiad-style questions love variable limits.

完整版本还能处理“上下限也依赖 a”的情况，需要额外加两个边界项。你应该记住完整版本，因为 IB Paper 3 和竞赛风格题目非常喜欢变限积分。

F(a) = ∫ᵤ₍ₐ₎^(v(a)) f(x, a) dx ⟹ F′(a) = ∫ᵤ^(v) (∂f/∂a) dx + f(v, a)·v′(a) − f(u, a)·u′(a)

Read the formula in three parts: the “interior” term where only the integrand changes, the top boundary term where the upper limit sweeps outward, and the bottom boundary term where the lower limit sweeps inward (hence the minus sign).

把公式读成三部分：第一部分是“内部”项，只有被积函数在变；第二部分是上限向外扫动带来的边界项；第三部分是下限变化带来的边界项（所以带负号）。

3. Why the Rule Works: A First-Principles Derivation | 原理推导：为什么公式成立

Start with the constant-limit case and write the definition of the derivative as a difference quotient. Let F(a) = ∫ f(x, a) dx. Then for a small increment h, we form the quotient and compare the two integrals over the same x-interval, which allows the integrals to be combined into a single one.

从常数上下限情形出发，用差商写出导数的定义。设 F(a) = ∫ f(x, a) dx。取一个小增量 h，构造差商，两个积分在同一 x 区间上，因此可以合并成一个积分。

[F(a + h) − F(a)] / h = ∫ [ f(x, a + h) − f(x, a) ] / h dx

Now let h → 0. Inside the integral, the fraction [f(x, a + h) − f(x, a)]/h is precisely the definition of the partial derivative ∂f/∂a at the point (x, a). If that convergence is well behaved for all x in the interval, the limit may be moved inside the integral sign, giving the rule.

令 h → 0。积分号内 [f(x, a + h) − f(x, a)]/h 正是点 (x, a) 处偏导数 ∂f

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