📚 IB Math: Anti-Differentiation and Finding the Original Function | IB数学：反微分与求原函数
Anti-differentiation is the reverse process of differentiation. Given a derivative, we try to recover the original function. In IB Mathematics AA and AI, this skill is essential for solving differential equations, computing areas, and modelling motion. This guide explains the core rules, notation, and problem-solving techniques with worked examples.
反微分是微分的逆过程。已知导函数，我们尝试还原出原函数。在 IB 数学 AA 与 AI 课程中，这项技能对于解微分方程、计算面积以及建立运动模型至关重要。本指南将通过例题讲解核心法则、记法和解题技巧。
1. What Is Anti-Differentiation? | 什么是反微分？
Differentiation maps a function f(x) to its derivative f'(x). Anti-differentiation maps f'(x) back to a family of functions whose derivative is f'(x). If F'(x) = f(x), then F is called an antiderivative (or primitive) of f.
微分将一个函数 f(x) 映为它的导函数 f'(x)。反微分则将 f'(x) 映回一族以 f'(x) 为导数的函数。若 F'(x) = f(x)，则称 F 是 f 的一个反导数（或原函数）。
If F'(x) = f(x), then ∫ f(x) dx = F(x) + C
若 F'(x) = f(x)，则 ∫ f(x) dx = F(x) + C
For example, since the derivative of x² is 2x, an antiderivative of 2x is x². However, the derivative of x² + 3 is also 2x, because the derivative of a constant is zero. This is why anti-differentiation produces a whole family of functions.
例如，因为 x² 的导数是 2x，所以 2x 的一个反导数是 x²。但 x² + 3 的导数同样也是 2x，因为常数的导数为零。这就是为什么反微分会得到一族函数。
2. The Constant of Integration | 积分常数
Since the derivative of any constant is zero, an antiderivative is never unique. The symbol C represents any real number and is called the constant of integration.
因为任何常数的导数都是零，所以反导数永远不唯一。符号 C 表示任意实数，称为积分常数。
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The derivative of x² is 2x, and so is the derivative of x² + 5 or x² – 100.
x² 的导数是 2x，x² + 5 或 x² – 100 的导数也都是 2x。
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Therefore ∫ 2x dx = x² + C, where C cannot be recovered from the derivative alone.
因此 ∫ 2x dx = x² + C，其中 C 无法仅从导函数还原出来。
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Extra information, such as an initial condition, is needed to find C.
需要额外的信息，例如初始条件，才能确定 C。
| Function f(x) | Antiderivative F(x) + C |
|---|---|
| 2x | x² + C |
| 2x + 7 | x² + 7x + C |
| 3x² | x³ + C |
3. The Power Rule | 幂法则
For any real number n ≠ -1, the power rule states:
对任意实数 n ≠ -1，幂法则为：
∫ xn dx = xn+1 / (n+1) + C
This rule works for positive powers, negative powers, and fractional powers, provided that n ≠ -1.
只要 n ≠ -1，该法则对正幂、负幂和分数幂均适用。
Example 1: ∫ x³ dx = x⁴ / 4 + C.
例 1：∫ x³ dx = x⁴ / 4 + C。
Example 2: ∫ √x dx = ∫ x1/2 dx = (2/3)x3/2 + C.
例 2：∫ √x dx = ∫ x1/2 dx = (2/3)x3/2 + C。
4. Integrating a Constant | 常数的积分
The antiderivative of a constant a is ax + C, because the derivative of ax is a.
常数 a 的反导数是 ax + C，因为 ax 的导数是 a。
∫ a dx = ax + C
Example: ∫ 5 dx = 5x + C.
例：∫ 5 dx = 5x + C。
5. Antiderivatives of Trigonometric Functions | 三角函数的反导数
The six standard trigonometric antiderivatives appear frequently in IB exams. Memorise them carefully.
以下六个标准三角反导数在 IB 考试中经常出现，请仔细记忆。
| Integral | Result |
|---|---|
| ∫ sin x dx | -cos x + C |
| ∫ cos x dx | sin x + C |
| ∫ sec² x dx | tan x + C |
| ∫ csc² x dx | -cot x + C |
| ∫ sec x tan x dx | sec x + C |
| ∫ csc x cot x dx | -csc x + C |
Notice the sign patterns: sin changes to -cos, and csc² changes to -cot.
注意符号规律：sin 变为 -cos，csc² 变为 -cot。
6. Exponential and Logarithmic Antiderivatives | 指数函数和对数函数的反导数
Exponential functions have a simple antiderivative because they are proportional to their own derivative.
指数函数的反导数非常简洁，因为它与自身的导数成正比。
∫ ex dx = ex + C
∫ ekx dx = (1/k) ekx + C, k ≠ 0
For the reciprocal function, the antiderivative uses the natural logarithm with absolute value:
对于倒数函数 1/x，其反导数需要用到带绝对值的自然对数：
∫ 1/x dx = ln|x| + C, x ≠ 0
Example: ∫ e3x dx = (1/3)e3x + C.
例：∫ e3x dx = (1/3)e3x + C。
7. Linearity Rules: Sums, Differences and Constant Multiples | 线性法则：和、差与常数倍
Anti-differentiation is linear. The integral of a sum is the sum of the integrals, and a constant factor can be pulled outside the integral sign.
反微分具有线性。和的积分等于积分的和，常数因子可以提到积分号外面。
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