📚 Indefinite Integrals (Antidifferentiation) — A Conceptual Guide | 不定积分（反微分）概念精讲
Indefinite integration, also known as antidifferentiation, is the reverse process of differentiation. While differentiation asks “what is the rate of change of this function?”, integration asks “which function could have produced this derivative?” This reversal is far more subtle than it first appears, and it lies at the heart of calculus.
不定积分，又称反微分，是微分的逆运算。微分问的是”这个函数的变化率是多少？”，而积分问的是”哪个函数求导后能得到这个导函数？”这种反向思维远比初看时更为微妙，也是整个微积分学的核心所在。
1. What Is an Antiderivative? | 什么是原函数？
Given a function f(x), an antiderivative is any function F(x) such that F'(x) = f(x). For example, if f(x) = 2x, then F(x) = x² is an antiderivative, because the derivative of x² is exactly 2x. However, F(x) = x² + 1, F(x) = x² − 5, and F(x) = x² + π are also antiderivatives, since the derivative of any constant is zero.
给定函数 f(x)，其原函数是满足 F'(x) = f(x) 的任意函数 F(x)。例如，若 f(x) = 2x，那么 F(x) = x² 就是一个原函数，因为 x² 的导数恰好是 2x。然而，F(x) = x² + 1、F(x) = x² − 5 以及 F(x) = x² + π 也都是原函数，因为任何常数的导数为零。
The key insight is that antidifferentiation is not unique. Once you find one antiderivative, you have actually found infinitely many, all differing only by a constant. This is why the most general form of an antiderivative is always written with a “+ C” term.
关键之处在于：反微分的结果并不唯一。一旦你找到一个原函数，实际上就找到了无穷多个，它们之间只相差一个常数。这就是为什么原函数的一般形式总是写成带”+ C”的表达式。
2. The Indefinite Integral Notation | 不定积分的记号
The symbol for integration is a stretched S: ∫. It was introduced by Leibniz in the 17th century, based on the Latin word “summa” (sum), because integration originally arose from summing infinitely many infinitesimally small pieces. The notation for the indefinite integral of f(x) with respect to x is:
积分符号是一个拉长的 S：∫。它由莱布尼茨在17世纪引入，源自拉丁词”summa”（和），因为积分最初源于将无穷多个无穷小量相加。函数 f(x) 关于 x 的不定积分记作：
∫ f(x)dx = F(x) + C, where F'(x) = f(x)
Here, ∫ is the integral sign, f(x) is the integrand, dx indicates that the integration is performed with respect to the variable x, and C is the constant of integration. The dx is not decorative — it tells us which variable is being integrated and must always be written.
其中，∫ 是积分号，f(x) 是被积函数，dx 表示对变量 x 进行积分，C 是积分常数。dx 并非装饰——它指明了积分变量，必须始终书写。
3. Why Is “+ C” Essential? | 为什么 “+ C” 必不可少？
Consider the antiderivative of f(x) = 3x². Since d/dx(x³) = 3x², one antiderivative is x³. But d/dx(x³ + 7) = 3x² too. If we were to write only ∫3x²dx = x³, we would be singling out one particular antiderivative while ignoring all others. The correct statement is:
考虑 f(x) = 3x² 的原函数。由于 d/dx(x³) = 3x²，一个原函数是 x³。但 d/dx(x³ + 7) = 3x² 同样成立。如果我们只写 ∫3x²dx = x³，就等于在无穷多个原函数中只挑了一个而忽略其他。正确的写法是：
∫ 3x² dx = x³ + C
The constant C represents any real number. It captures the family of all possible antiderivatives. Graphically, these functions are vertical translations of one another — identical curves shifted up or down along the y-axis. Every point on the curve x³ + C shares the same slope pattern, meaning the derivative at each x-value is the same for all C.
常数 C 代表任何实数。它囊括了全体原函数构成的函数族。从图形上看，这些函数互为纵坐标平移——是沿 y 轴上移或下移的相同曲线。曲线 x³ + C 上每一点的斜率模式相同，即对于所有 C，同一 x 值处的导数相同。
4. Basic Integration Rules for Power Functions | 幂函数的基本积分法则
For n ≠ −1, the power rule of antidifferentiation says that to integrate xⁿ, we increase the exponent by 1 and divide by the new exponent:
当 n ≠ −1 时，幂函数的反微分法则指出：对 xⁿ 积分，需将指数加 1，再除以新的指数：
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C (n ≠ −1)
Let’s verify with n = 2: ∫ x² dx = x³/3 + C. Differentiating x³/3 gives 3x²/3 = x². ✓ The condition n ≠ −1 is critical, because if n = −1, the formula would require division by zero. In that case, the rule is replaced by a special formula:
以 n = 2 验证：∫ x² dx = x³/3 + C。对 x³/3 求导得 3x²/3 = x²。✓ 条件 n ≠ −1 至关重要，因为若 n = −1，公式中就会出现除以零。此时，法则由一条特殊公式替代：
∫ x⁻¹ dx = ∫ 1/x dx = ln|x| + C (x ≠ 0)
The absolute value sign is essential because ln x is only defined for x > 0, but 1/x is defined for both negative and positive x. Since d/dx(ln|x|) = 1/x for x ≠ 0, this formula covers the entire domain of the integrand.
绝对值符号必不可少，因为 ln x 仅在 x > 0 时有定义，而 1/x 对正负数都有定义。由于 x ≠ 0 时 d/dx(ln|x|) = 1/x，此公式覆盖了被积函数的完整定义域。
5. Common Basic Integrals You Must Know | 必须掌握的基本积分公式
The following table lists the most frequently used antiderivatives in the IB Mathematics curriculum. Each formula can be verified by differentiating the right-hand side.
下表列出了 IB 数学课程中最常用的原函数公式。每一个都可通过右侧求导来验证。
| ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C | n ≠ −1 | 幂函数积分（n 元） |
| ∫ 1/x dx = ln|x| + C | x ≠ 0 | 倒数函数积分（n = −1 特例） |
| ∫ sin x dx = −cos x + C | — | 正弦函数积分 |
| ∫ cos x dx = sin x + C | — | 余弦函数积分 |
| ∫ eˣ dx = eˣ + C | — | 指数函数积分 |
| ∫ aˣ dx = aˣ/ln a + C | a > 0, a ≠ 1 | 一般指数函数积分 |
| ∫ sec² x dx = tan x + C | — | sec² 函数积分 |
| ∫ 1/(1+x²) dx = arctan x + C | — | 反正切积分 |
| ∫ 1/√(1−x²) dx = arcsin x + C | |x| < 1 | 反正弦积分 |
Memorising these formulas is essential, but understanding how they are derived from differentiation rules is equally important. Every integration formula is simply a derivative rule read in reverse.
熟记这些公式必不可少，但理解它们如何由微分法则推导而来同样重要。每一条积分公式无非是一条微分法则的倒读。
6. Linearity: Sums and Constant Multiples | 线性性质：加法与常数倍
Indefinite integration is a linear operation, which means it respects addition and scalar multiplication. For any functions f(x) and g(x), and any constant k:
不定积分是线性运算，意味着它满足加法与数乘的运算规则。对于任意函数 f(x) 与 g(x)，以及任意常数 k：
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
∫ k·f(x) dx = k·∫ f(x) dx
These properties allow us to integrate complicated expressions term by term. For instance:
这些性质使我们能够逐项积分复杂的表达式。例如：
∫ (4x³ − 2x + 5) dx = 4·(x⁴/4) − 2·(x²/2) + 5x + C = x⁴ − x² + 5x + C
Notice that we only write one single C at the end, even though each term contributes an arbitrary constant. The sum of several constants is still just one arbitrary constant, so it would be redundant to write C₁ + C₂ + C₃.
注意：即使每一项都贡献一个任意常数，我们最终也只写一个 C。因为多个常数之和仍然只是一个任意常数，分别写 C₁ + C₂ + C₃ 是多余的。
7. Integration by Substitution: The Reverse Chain Rule | 换元积分法：链式法则的逆运算
When the integrand looks like a composite function multiplied by the derivative of its inner function, substitution is the natural technique. Consider ∫ 2x·cos(x²) dx. Notice that d/dx(x²) = 2x, which is exactly the other factor present. Let u = x², so du/dx = 2x, which we write formally as du = 2x dx. Then the integral becomes simplicity itself:
当被积函数形如复合函数乘以内层函数的导数时，换元法就是自然的选择。考虑 ∫ 2x·cos(x²) dx。注意 d/dx(x²) = 2x，正好是另一个因子。令 u = x²，则 du/dx = 2x，形式地写作 du = 2x dx。于是积分变得极其简单：
∫ 2x·cos(x²) dx = ∫ cos u du = sin u + C = sin(x²) + C
The fully worked procedure follows these four steps:
完整的操作步骤如下四步：
-
Choose a substitution u = g(x) where g'(x) appears (up to a constant factor) in the integrand. | 选择一个替换 u = g(x)，使 g'(x)（至多差一个常数倍数）出现在被积函数中。
-
Compute du = g'(x) dx and rewrite the entire integral in terms of u. | 计算 du = g'(x) dx，并将整个积分改写为关于 u 的表达式。
-
Integrate with respect to u using the basic formulas. | 使用基本公式对 u 积分。
-
Substitute back u = g(x) to express the answer in terms of the original variable. | 代回 u = g(x)，将答案表达为原变量 x 的函数。
8. Integration by Parts: The Reverse Product Rule | 分部积分法：乘积法则的逆运算
When the integrand is a product of two unrelated functions, such as x·eˣ or x·sin x, neither substitution nor direct formulas suffice. The technique of integration by parts is derived from the product rule. If u and v are functions of x, then the product rule states:
当被积函数是两个无关函数的乘积（如 x·eˣ 或 x·sin x）时，置换法与基本公式都不够用。分部积分法源自乘积法则。若 u 与 v 均为 x 的函数，乘积法则指出：
d/dx(uv) = u(dv/dx) + v(du/dx)
Integrating both sides and rearranging gives the integration by parts formula:
两边积分并整理，得到分部积分公式：
∫ u dv = uv − ∫ v du
For ∫ x·eˣ dx, choose u = x and dv = eˣ dx. Then du = dx and v = eˣ. Applying the formula:
对于 ∫ x·eˣ dx，取 u = x，dv = eˣ dx。则 du = dx，v = eˣ。代入公式：
∫ x·eˣ dx = x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C
The key is choosing u to be the function that simplifies upon differentiation (like x), and dv to be the function that does not become more complicated upon integration (like eˣ or sin x).
关键在于选择 u 为求导后能简化的函数（如 x），dv 为积分后不会变复杂的函数（如 eˣ 或 sin x）。
9. Initial Conditions: Finding the Specific Antiderivative | 初始条件：确定具体的原函数
While the indefinite integral is written with a general constant C, we can determine C if we are given one additional piece of information, such as a point that the curve passes through. This is known as an initial condition or boundary condition.
不定积分虽然写成带通用常数 C，但若已知一个附加条件（例如曲线经过某个点），就能确定 C。这个条件称为初始条件或边界条件。
Suppose we know that ∫ f(x) dx = x³ + 2x + C, and we are told that F(1) = 5. Then:
假设已知 ∫ f(x) dx = x³ + 2x + C，并给出条件 F(1) = 5。则：
F(1) = 1³ + 2(1) + C = 3 + C = 5 ⇒ C = 2
Thus the specific antiderivative is F(x) = x³ + 2x + 2. This procedure is identical to the one used to find the constant terms when solving differential equations. In the IB syllabus, this skill is tested in both the Analysis and Approaches (AA) and Applications and Interpretation (AI) courses.
于是具体原函数为 F(x) = x³ + 2x + 2。此方法在解微分方程时用来求常数项，在 IB 的 Analysis and Approaches（AA）与 Applications and Interpretation（AI）课程中均有涉及。
10. Common Mistakes and How to Avoid Them | 常见错误及避免策略
One of the most common errors is forgetting the “+ C” in an indefinite integral. In a definite integral, the constant cancels out, so it is not required. But for an indefinite integral, omitting C means the answer is incomplete — it fails to represent the entire family of antiderivatives. IB examiners consistently penalise this omission.
最常见的错误之一是忘记在不定积分中写”+ C”。在定积分中，常数会相消，因此不必写出。但对于不定积分，漏写 C 意味着答案不完整——未能表示全部原函数族。IB 考官对此扣分是常态。
Another frequent error is misapplying the power rule when n = −1. The formula ∫ xⁿ dx requires division by n + 1, which becomes invalid at n = −1. Many students incorrectly write ∫ x⁻¹ dx = x⁰/0, which is mathematically meaningless. The correct result is ln|x| + C.
另一个常见错误是在 n = −1 时误用幂函数法则。公式 ∫ xⁿ dx 需要除以 n + 1，当 n = −1 时分母为零。很多学生错误地写出 ∫ x⁻¹ dx = x⁰/0，这在数学上毫无意义。正确结果是 ln|x| + C。
Finally, students often forget to add the absolute value in ∫ 1/x dx = ln|x| + C. Without the absolute value, the formula is only valid for positive x. Including it ensures correctness over the full domain where 1/x is defined.
最后，学生常在 ∫ 1/x dx = ln|x| + C 中漏写绝对值。没有绝对值符号，该公式仅对正数 x 成立。加上它才能确保公式在 1/x 的完整定义域上成立。
11. Verifying Your Antiderivative | 验证原函数的方法
Because antidifferentiation is the inverse of differentiation, every answer can be checked by differentiating. This is a powerful self-checking tool that students should use before submitting any answer. If the derivative of your proposed antiderivative equals the original integrand, then the answer is correct.
因为反微分是微分的逆运算，每个答案都可以通过求导来检验。这是学生在提交答案之前应当使用的强力自检工具。如果你提出的原函数的导数恰好等于原被积函数，那么这个答案就是正确的。
For example, to verify ∫ (6x⁵ − 4x³ + 8) dx = x⁶ − x⁴ + 8x + C, differentiate the right side:
例如，验证 ∫ (6x⁵ − 4x³ + 8) dx = x⁶ − x⁴ + 8x + C，只需对右侧求导：
d/dx(x⁶ − x⁴ + 8x + C) = 6x⁵ − 4x³ + 8 ✓
If the derivative matches the integrand, the integration is verified. Note that the constant C differentiates to zero, so it never affects the verification process.
如果导数与被积函数一致，则积分得到验证。注意常数 C 的导数为零，因此在验证过程中它从不产生任何影响。
12. Connection to Definite Integrals | 与定积分的联系
Indefinite integrals and definite integrals are closely linked through the Fundamental Theorem of Calculus. The indefinite integral represents a family of functions, while the definite integral produces a numerical value representing the signed area under a curve. When evaluating a definite integral ∫ₐᵇ f(x)dx, we use the antiderivative F(x) and compute F(b) − F(a). The constant C conveniently cancels out:
不定积分与定积分通过微积分基本定理紧密相连。不定积分表示一族函数，而定积分产生一个数值，代表曲线下的有向面积。在计算定积分 ∫ₐᵇ f(x)dx 时，我们使用原函数 F(x) 并计算 F(b) − F(a)。常数 C 恰好相消：
∫ₐᵇ f(x)dx = [F(x) + C]ₐᵇ = [F(b) + C] − [F(a) + C] = F(b) − F(a)
This is why, when only the indefinite integral is requested, writing “+ C” is mandatory; but when evaluating a definite integral, C disappears automatically. Understanding this relationship deepens your grasp of both concepts and prevents the common confusion between the “family of functions” interpretation and the “signed area” interpretation.
这就是为什么当仅要求不定积分时写”+ C”是必须的；而在计算定积分时 C 自动消失。理解这一关系有助于同时深化对两个概念的理解，并避免将”函数族”与”有向面积”两种解释相混淆。
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