📚 Antidifferentiation | 反微分法
tegration, is the process of finding a function whose derivative is a given function. If F' C, where C is an arbitrary constant. This topic is foundational in IB Mathematics, linking directly to area calculations, differential equations, and motion problems.
反微分，常称为不定积分，是寻找一个函数使其导数等于给定函数的过程。如果 F'(x) = f(x)，那么 F 就是 f 的一个反导数。所有反导数的集合写作 ∫ f(x) dx = F(x) + C，其中 C 为任意常数。这一主题是 IB 数学的基础，直接联系面积计算、微分方程和运动问题。
1. Definition of Antiderivative | 反导数的定义
A function F(x) is called an antiderivative of f(x) on an interval if F' nce the derivative of x² is 2x, an antiderivative of 2x is x². However, note that (x² + 5)' = 2x as well, so antiderivatives are not unique; they differ by a constant.
如果在某个区间上 F'(x) = f(x) 恒成立，则称 F(x) 是 f(x) 的一个反导数。例如，x² 的导数是 2x，因此 2x 的一个反导数是 x²。但注意到 (x² + 5)' = 2x 同样成立，所以反导数并不唯一，它们之间相差一个常数。
2. The Indefinite Integral and Constant of Integration | 不定积分与积分常数
gral sign, f(x) is the integrand, and dx indicates the variable of integration. The constant C emerges because differentiation eliminates constant terms. Without +C, the solution set is incomplete.
记号 ∫ f(x) dx 表示 f 关于 x 的不定积分，给出了所有的反导数族。符号 ∫ 是积分号，f(x) 是被积函数，dx 表明积分变量。常数 C 的出现是因为微分会消去常数项。缺少 +C 将使解集不完整。
3. Basic Integration Rules | 基本积分法则
Just as differentiation has standard rules, integration can be approached with a set of fundamental formulas derived from reversing differentiation. The table below summarises the key rules; each can be verified by differentiating the right-hand side.
如同微分有标准规则，积分也可以用一组从微分逆推得到的基本公式来处理。下表总结了关键法则，每一项都可以通过对右边求导来验证。
| Rule / 规则 | Formula / 公式 |
|---|---|
| Constant Rule | ∫ k dx = kx + C |
| Power Rule (n ≠ -1) | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C |
| Sum/Difference Rule | ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx |
| Constant Multiple Rule | ∫ k f(x) dx = k ∫ f(x) dx |
这些规则是积分运算的基石。特别是幂法则，要求 n ≠ -1，因为当 n = -1 时，会得到 ∫ x⁻¹ dx = ln|x| + C，这属于对数情形的特殊情况。常数乘法和加减法则允许我们将复杂积分拆分为简单积分的线性组合，极大简化计算。
4. The Power Rule in Detail | 幂函数积分详解
For any real number n except -1, the antiderivative of xⁿ is obtained by increasing the exponent by 1 and dividing by the new exponent. In symbols: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. For instance, ∫ x³ dx = x⁴/4 + C and ∫ √x dx = ∫ x½ dx = x³⁄²/(3/2) + C = (2/3)x³⁄² + C.
对于任何不等于 -1 的实数 n，xⁿ 的反导数是把指数加 1 再除以新的指数。符号表示为：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。例如，∫ x³ dx = x⁴/4 + C；∫ √x dx = ∫ x½ dx = x³⁄²/(3/2) + C = (2/3)x³⁄² + C。
x + C. The exception n = -1 leads to the natural logarithm, which will be discussed later.
在积分前，务必把根式和分式写成指数形式。例如 ∫ 1/x² dx = ∫ x⁻² dx = x⁻¹/(-1) + C = -1/x + C。n = -1 的例外导致自然对数，这将在后面讨论。
5. Integrals of Trigonometric Functions | 三角函数的积分
∫ csc² x dx = -cot x + C, ∫ sec x tan x dx = sec x + C, and ∫ csc x cot x dx = -csc x + C.
标准三角函数的积分可以通过逆向它们的导数来记忆。主要结果有：∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C。对于平方函数，∫ 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。
These formulas work when the angle is simply x. When the integrand involves a linear function of x, e.g. sin(2x), the reverse chain rule is needed, which is covered under substitution. Many IB problems combine trigonometric integrals with the power rule after simplification.
这些公式在角度仅为 x 时有效。当被积函数包含 x 的线性函数时，如 sin(2x)，就需要用到逆链式法则，这会在代换法部分讲解。许多 IB 题目在化简后将三角函数积分与幂法则结合使用。
6. Exponential and Logarithmic Integrals | 指数与对数积分
The exponential function eˣ is its own derivative, therefore ∫ eˣ dx = eˣ + C. For a general base a > 0, a ≠ 1, ∫ aˣ dx = aˣ/ln a + C. The natural logarithm fills the gap left by the power rule: ∫ 1/x dx = ln|x| e the absolute value ensures the domain matches the antiderivative for negative x as well.
指数函数 eˣ 的导数就是它本身，因此 ∫ eˣ dx = eˣ + C。对于一般底数 a > 0, a ≠ 1，∫ aˣ dx = aˣ/ln a + C。自然对数填补了幂法则留下的空白：∫ 1/x dx = ln|x| + C，其中的绝对值确保反导数在负数 x 上也有定义。
A particularly useful formula arises from the derivative of ln|f(x)|: ∫ f'(x)/f(x) dx = ln|f(x)| integrals of rational functions where the numerator is the derivative of the denominator.
一个特别有用的公式来自 ln|f(x)| 的导数：∫ f'(x)/f(x) dx = ln|f(x)| + C。识别出这一模式可以大大简化那些分子恰好是分母导数的有理函数的积分。
7. Integration by Substitution (Reverse Chain Rule) | 换元积分法（逆链式法则）
the inner function appears (up to a constant factor). Formally, let u = g(x), then du = g'(x) dx, and ∫ f(g(x)) g'(x) dx = ∫ f(u) du.
换元法是链式法则在积分中的对应。当被积函数包含一个复合函数，并且内部函数的导数（最多差一个常数倍）出现时，就可以使用换元法。正式地，令 u = g(x)，则 du = g'(x) dx，于是 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。
tion is a core technique in IB exams, often tested with trigonometric and exponential functions.
例如，计算 ∫ 2x (x² + 3)⁴ dx。令 u = x² + 3，则 du = 2x dx，得到 ∫ u⁴ du = u⁵/5 + C = (x²+3)⁵/5 + C。换元法是 IB 考试的核心技巧，常与三角函数和指数函数结合考查。
8. Recognizing Derivatives from Standard Functions | 从标准函数识别导数
s exactly the derivative of a known function. Familiar patterns include: ∫ (1/(x ln a)) dx? Actually, the derivative of logₐ x is 1/(x ln a), so ∫ 1/(x ln a) dx = logₐ|x| + C. More commonly, ∫ 1/(1+x²) dx = arctan x + C, ∫ 1/√1-x² dx = arcsin x + C.
许多积分可以通过识别被积函数恰好是某个已知函数的导数来求解。熟悉的模式有：1/(x ln a) 的反导数是 logₐ|x|；更常见的是，∫ 1/(1+x²) dx = arctan x + C，∫ 1/√1-x² dx = arcsin x + C。
These inverse trigonometric integrals appear in IB Analysis & Approaches and can also be approached using trigonometric substitution. Knowing the derivatives of arcsin, arctan and arcsec allows quick integration of rational and radical forms that might otherwise require lengthy work.
这些反三角函数的积分在 IB 分析与方法中出现，也可以通过三角代换处理。记住 arcsin、arctan 和 arcsec 的导数能够快速积分某些有理式和根式，否则可能需要冗长的计算。
9. Initial Conditions and Finding the Particular Solution | 初始条件与求特解
nt on the graph of the antiderivative, the constant C can be determined. For example, if f' ing x=1 gives 1³ + C = 4 => C = 3, so f(x) = x³ + 3. This is called a particular solution.
如果给出附加信息，比如反导数图形上的一个点，就可以确定常数 C。例如，已知 f'(x) = 3x² 且 f(1) = 4，则 f(x) = x³ + C。代入 x=1 得 1 + C = 4，故 C = 3，因此 f(x) = x³ + 3。这称为特解。
sition or velocity helps to pin down the constants and model the physical situation accurately.
在 IB 问题中，初始条件通常在速度和加速度的背景下给出：v(t) = ∫ a(t) dt，s(t) = ∫ v(t) dt。利用初始位置或速度可以确定常数，从而准确建立物理情境的模型。
10. Definite vs Indefinite Integrals | 定积分与不定积分对比
of Calculus connects them: ∫ₐᵇ f(x) dx = F(b) - F(a), where F is any antiderivative of f.
不定积分表示带常数 C 的函数族，而定积分 ∫ₐᵇ f(x) dx 则计算一个实数，代表曲线与 x 轴之间的净面积。微积分基本定理将它们联系起来：∫ₐᵇ f(x) dx = F(b) - F(a)，其中 F 是 f 的任意一个反导数。
Understanding the distinction is crucial. When evaluating a definite integral, never include +C; instead, substitute the limits into an antiderivative and subtract. IB examiners often mark incorrect inclusion of C in definite integrals as a conceptual error.
理解这一区别至关重要。计算定积分时，不要加 +C，而是将上下限代入一个反导数并相减。IB 考官常将在定积分中加入 C 视为概念性错误。
11. Common Mistakes and Misconceptions | 常见错误与误解
ct as the product of integrals, which is incorrect: ∫ f(x)g(x) dx ≠ ∫ f(x) dx · ∫ g(x) dx.
一个常见错误是在不定积分中忘记积分常数。另一个是在 n = -1 时误用幂法则，导致除以零。学生有时还会把积分的乘积当作乘积的积分，这是错误的：∫ f(x)g(x) dx ≠ ∫ f(x) dx · ∫ g(x) dx。
Other pitfalls include: forgetting to adjust dx when substituting, dropping absolute values in ln|f(x)|, and assuming all functions can be integrated in elementary terms. IB problems often test these subtle points, so careful verification by differentiation is strongly recommended.
其他陷阱包括：换元时忘记调整 dx，省略 ln|f(x)| 中的绝对值，以及认为所有函数都能用初等函数积分。IB 题目经常考查这些细微之处，因此强烈建议通过微分验证结果。
12. IB Exam Tips | IB 考试技巧
Always show the constant of integration in indefinite integrals unless a particular solution is requested. When using substitution, write down u and du explicitly. Check your answer by differentiating mentally; this can catch sign errors and forgotten constants. For complicated rational functions, look for the f'(x)/f(x) pattern before attempting polynomial division.
除非要求特解，否则不定积分中务必写出积分常数。使用换元法时，明确写出 u 和 du。通过心算微分来检查答案，这能发现符号错误和遗忘的常数。对于复杂的有理函数，先寻找 f'(x)/f(x) 模式，再考虑多项式除法。
In IB Analysis & Approaches, be comfortable with integrals involving eˣ, ln x, sin, cos, and their combinations. In Applications & Interpretation, indefinite integrals may appear in the context of marginal cost, velocity, and probability density functions. Practise recognising the reverse chain rule without explicit substitution to save time.
在 IB 分析与方法中，要熟悉涉及 eˣ、ln x、sin、cos 及其组合的积分。在应用与解释中，不定积分可能出现在边际成本、速度和概率密度函数的情境中。练习在不明确换元的情况下识别逆链式法则，以节省时间。
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