From Derivatives to Antiderivatives: The Basic Idea of Reversing Differentiation | 从导数到原函数：逆转微分的基本思路

📚 From Derivatives to Antiderivatives: The Basic Idea of Reversing Differentiation | 从导数到原函数：逆转微分的基本思路

Differentiation is a powerful mathematical machine: it takes a function and produces another function that describes rates of change. But calculus also asks the reverse question. If we know the derivative f(x), can we recover a function F(x) whose derivative is f(x)? This reverse process is called antidifferentiation, and its result is called an antiderivative or indefinite integral. In this article, we will develop the basic idea of reversing differentiation from the ground up.

微分是一台强大的数学机器：输入一个函数，它输出另一个描述变化率的函数。但微积分还要追问反向问题：如果我们知道导数 f(x)，能否找回一个函数 F(x)，使得它的导数正好是 f(x)？这个逆向过程称为反微分（求原函数），其结果称为原函数或不定积分。本文将从最基础的思想开始，讲清楚如何逆转微分。

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1. What Does It Mean to Reverse Differentiation? | 逆转微分是什么意思？

To reverse differentiation means to start with a known derivative and find at least one function that could have produced it. For example, the derivative of x² is 2x, so one antiderivative of 2x is x². But because the derivative of a constant is zero, x²+1, x²−7, and x²+C all also have derivative 2x. Reversing differentiation is therefore not a simple one-step operation; it produces a whole family of possible functions.

逆转微分，就是从一个已知导数出发，去寻找至少一个能够产生它的函数。例如，x² 的导数是 2x，所以 2x 的一个原函数是 x²。但由于常数的导数为零，x²+1、x²−7 以及一切 x²+C 的导数也都是 2x。因此，逆转微分并不是一步到位的简单运算，它会产生一族可能的函数。

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2. The Definition of an Antiderivative | 原函数的定义

Formally, a function F is called an antiderivative of f on an interval I if F′(x)=f(x) for every x in I. The standard notation for the general antiderivative is the indefinite integral:

严格地说，如果对于区间 I 上每一个 x，都有 F′(x)=f(x)，则称 F 是 f 在 I 上的一个原函数。原函数族的标准记号就是不定积分：

∫ f(x) dx = F(x) + C，其中 F′(x) = f(x)

The symbol ∫ is an elongated S, suggesting summation, and dx tells us that the variable of integration is x. The constant C is called the constant of integration. In IB mathematics, the terms antiderivative and indefinite integral are often used interchangeably.

积分号 ∫ 是一个拉长的 S，暗示求和；dx 表示积分变量是 x。常数 C 称为积分常数。在 IB 数学中，antiderivative 与 indefinite integral 常常可以互换使用。

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3. Why the Constant of Integration Matters | 为什么积分常数如此关键

Since the derivative of any constant is zero, adding a constant to a known antiderivative does not change its derivative. If F′(x)=f(x), then also (F(x)+C)′=f(x). This explains why every indefinite integral must include +C. Geometrically, the family F(x)+C represents a set of curves that are vertical translations of one another; at every fixed x-value, all these curves have the same slope.

因为任意常数的导数都是零，所以在一个已知原函数上加上常数，导数不会改变。如果 F′(x)=f(x)，那么也有 (F(x)+C)′=f(x)。这就解释了为什么每个不定积分都必须写 +C。从几何上看，F(x)+C 表示一族相互之间仅作上下平移的曲线；在同一个 x 处，所有这些曲线的斜率都相同。

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4. Basic Antiderivative Rules from Derivative Rules | 由导数规则推导基本积分规则

Every derivative rule can be read in reverse to obtain an antiderivative rule. If we have memorised the standard derivatives, we already know the answers to many indefinite integrals. The table below lists some of the most important reversals.

每一条导数公式都可以倒过来读，从而得到一条原函数公式。如果我们已经记住了常用导数表，那么很多不定积分的结果其实已经掌握了。下表列出一些最重要的逆转关系。

f(x)	∫ f(x) dx
k	kx + C
xⁿ (n≠−1)	xⁿ⁺¹/(n+1) + C
1/x	ln|x| + C
eˣ	eˣ + C
sin x	−cos x + C
cos x	sin x + C
sec²x	tan x + C

These formulas are direct reversals of familiar derivatives such as d/dx(xⁿ)=nxⁿ⁻¹ and d/dx(sin x)=cos x. It is essential to be able to recall them quickly, because almost every more complicated integral is built from these basic pieces.

这些公式是常见导数如 d/dx(xⁿ)=nxⁿ⁻¹、d/dx(sin x)=cos x 的直接逆转。我们必须能够快速想起这些结果，因为几乎所有更复杂的积分都是由这些基本部分组合而成的。

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5. The Power Rule in Reverse | 幂函数积分规则的逆转

For any power xⁿ with n≠−1, the antiderivative rule is:

对于任何幂函数 xⁿ，只要 n≠−1，原函数规则为：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

This rule is easy to verify: differentiating xⁿ⁺¹/(n+1) gives (n+1)xⁿ/(n+1)=xⁿ. For example, ∫ x² dx = x³/3 + C, and ∫ x⁻³ dx = x⁻²/(−2)+C = −1/(2x²)+C.

这条规则很容易验证：对 xⁿ⁺¹/(n+1) 求导，得到 (n+1)xⁿ/(n+1)=xⁿ。例如，∫ x² dx = x³/3 + C，又比如 ∫ x⁻³ dx = x⁻²/(−2)+C = −1/(2x²)+C。

The single exception is n=−1. The formula would require division by zero, so x⁻¹ is handled separately. Since d/dx(ln|x|)=1/x, we have:

唯一例外是 n=−1。此时公式中会出现除以零，所以 x⁻¹ 需要单独处理。因为 d/dx(ln|x|)=1/x，所以：

∫ x⁻¹ dx = ∫ 1/x dx = ln|x| + C

The absolute value is needed because ln x is only defined for x>0, while 1/x is defined for all x≠0.

绝对值符号是必要的，因为 ln x 只在 x>0 时有定义，而 1/x 对所有 x≠0 都有定义。

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6. Trigonometric Antiderivatives | 三角函数的原函数

The derivatives of trigonometric functions give us an immediate list of antiderivatives. Because d/dx(sin x)=cos x, we know that ∫ cos x dx = sin x + C. Because d/dx(cos x)=−sin x, we know that ∫ sin x dx = −cos x + C. Similarly, d/dx(tan x)=sec²x, so ∫ sec²x dx = tan x + C.

三角函数的导数直接给了我们一张原函数清单。因为 d/dx(sin x)=cos x，所以 ∫ cos x dx = sin x + C。因为 d/dx(cos x)=−sin x，所以 ∫ sin x dx = −cos x + C。同理，d/dx(tan x)=sec²x，所以 ∫ sec²x dx = tan x + C。

The most common sign error is to write ∫ sin x dx = cos x + C. This is wrong. The derivative of cos x is −sin x, so the correct antiderivative of sin x is −cos x + C. Always check the sign by differentiating your answer.

最常见的符号错误是把 ∫ sin x dx 写成 cos x + C。这是不对的。因为 cos x 的导数是 −sin x，所以 sin x 的正确原函数是 −cos x + C。一定要通过对答案求导来检查符号。

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7. Exponential and Logarithmic Antiderivatives | 指数与对数函数的原函数

The exponential function eˣ is particularly elegant because it is its own derivative and its own antiderivative:

指数函数 eˣ 非常优雅，因为它是自己的导数，也是自己的原函数：

∫ eˣ dx = eˣ + C

For a general exponential function aˣ with a>0 and a≠1, we use the derivative rule d/dx(aˣ)=aˣ ln a. Reversing it gives ∫ aˣ dx = aˣ/ln a + C. Meanwhile, the antiderivative of 1/x is logarithmic, as already noted: ∫ 1/x dx = ln|x|+C.

对于一般的指数函数 aˣ，其中 a>0 且 a≠1，利用导数公式 d/dx(aˣ)=aˣ ln a。倒过来就得到 ∫ aˣ dx = aˣ/ln a + C。同时，1/x 的原函数是对数函数，如前所述：∫ 1/x dx = ln|x|+C。

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8. Linearity: Adding and Scaling Antiderivatives | 线性性质：加法与常数倍

Antidifferentiation is a linear operation. This means that the integral of a sum is the sum of the integrals, and a constant factor can be pulled outside the integral sign:

反微分是线性运算。也就是说，和的积分等于积分的和，常数因子可以提到积分号外面：

∫ [a f(x) + b g(x)] dx = a∫ f(x) dx + b∫ g(x) dx

For example, ∫(3x²+4cos x) dx = 3∫x² dx + 4∫cos x dx = x³ + 4 sin x + C. This rule is extremely useful because it allows us to integrate term by term.

例如，∫(3x²+4cos x) dx = 3∫x² dx + 4∫cos x dx = x³ + 4 sin x + C。这条性质非常有用，因为它允许我们逐项积分。

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9. Checking Your Answer by Differentiation | 用微分检验答案

Once you compute an antiderivative, the most reliable way to check it is to differentiate your answer. If you return to

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