📚 Rules for Integration | 积分法则
Integration, the reverse process of differentiation, is a cornerstone of calculus and a vital component of the IB Mathematics syllabus. Mastering the rules for integration empowers you to solve a wide range of problems, from finding areas under curves to solving differential equations. This article provides a comprehensive guide to the essential integration techniques, from basic antiderivatives to advanced methods such as integration by parts and partial fractions, ensuring you are well-prepared for your IB exams.
积分，作为微分的逆运算，是微积分的基石，也是 IB 数学课程的重要组成部分。掌握积分法则能够帮助你解决广泛的问题，从计算曲线下的面积到求解微分方程。本文提供了一份全面的基本积分技巧指南，涵盖从基础反导数到分部积分法和部分分式等高级方法，确保你为 IB 考试做好充分准备。
1. Power Rule and Basic Antiderivatives | 幂法则与基本反导数
The most fundamental rule of integration is the power rule. For any real number n ≠ −1, the antiderivative of xⁿ is given by a simple formula where you increase the exponent by one and divide by the new exponent. Do not forget the constant of integration C.
最基本的积分法则是幂法则。对于任何实数 n ≠ −1，xⁿ 的反导数由一个简单的公式给出：指数加 1，然后除以新的指数。别忘了积分常数 C。
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ −1
When n equals −1, the formula breaks down because division by zero occurs. In that special case, the integral leads to the natural logarithm of the absolute value of x.
当 n = −1 时，该公式失效，因为分母为零。在这种特殊情况下，积分结果为 x 的绝对值的自然对数。
∫ x⁻¹ dx = ∫ 1/x dx = ln|x| + C
For constant functions, the integral is simply the constant multiplied by x. For example, ∫ 5 dx = 5x + C. The power rule can also handle roots by rewriting them as fractional exponents: ∫ √x dx = ∫ x¹⁄² dx = (2/3)x³⁄² + C.
对于常数函数，积分就是该常数乘以 x。例如，∫ 5 dx = 5x + C。幂法则还可以通过将根式写成分数指数来处理：∫ √x dx = ∫ x¹⁄² dx = (2/3)x³⁄² + C。
2. Constant Multiple and Sum/Difference Rules | 常数倍与和差法则
Integration is a linear operator, meaning you can pull constant factors out of the integral and integrate sums or differences term by term. This allows complex expressions to be broken down into simpler parts.
积分是一种线性运算，这意味着你可以把常数因子提到积分号外，并逐项积分和或差。这样可以将复杂表达式分解为更简单的部分。
∫ k·f(x) dx = k ∫ f(x) dx
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
For instance, to integrate 3x² − 4x + 2, you can integrate each term separately: 3∫ x² dx − 4∫ x dx + ∫ 2 dx. Applying the power rule then gives x³ − 2x² + 2x + C. Always remember that the constant C appears only once for the whole indefinite integral, not per term.
例如，要积分 3x² − 4x + 2，你可以分别积分每一项：3∫ x² dx − 4∫ x dx + ∫ 2 dx。应用幂法则得到 x³ − 2x² + 2x + C。请始终记住，常数 C 只在整个不定积分中写一次，而不是每项都写。
3. Integrals of Exponential and Logarithmic Functions | 指数函数与对数函数的积分
Exponential functions are notable because their integrals are closely related to the original function. The natural exponential function eˣ is particularly elegant, being its own antiderivative. For bases other than e, a simple division by the natural logarithm of the base is required.
指数函数值得注意，因为它们的积分与原函数密切相关。自然指数函数 eˣ 特别优美，它是自身的反导数。对于其他底数，只需除以底数的自然对数。
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ / ln a + C, a > 0, a ≠ 1
The integral of the natural logarithm function is not as immediately obvious and is typically derived using integration by parts. Its formula is a standard result you must know.
自然对数函数的积分并非一目了然，通常通过分部积分法推导。其公式是必须掌握的标准结果。
∫ ln x dx = x ln x − x + C
In IB problems, you will often need to integrate expressions like e³ˣ by using substitution. For example, ∫ e³ˣ dx = (1/3)e³ˣ + C. Similarly, the integral of 1/(x) or any fraction where the numerator is the derivative of the denominator leads to ln|denominator| + C. This pattern is extremely useful: ∫ (g'(x)/g(x)) dx = ln|g(x)| + C.
在 IB 考题中，常需要通过代换来积分像 e³ˣ 这样的表达式。例如，∫ e³ˣ dx = (1/3)e³ˣ + C。同理，1/x 或分子为分母导数的任何分式的积分结果都是 ln|分母| + C。这一模式极其有用：∫ (g'(x)/g(x)) dx = ln|g(x)| + C。
4. Integrals of Trigonometric Functions | 三角函数的积分
Knowing the antiderivatives of the six basic trigonometric functions by heart is essential. These come directly from reversing differentiation rules. Pay special attention to the signs, as negative signs are a common source of errors.
熟记六个基本三角函数的反导数至关重要。它们直接来自微分法则的逆转。要特别注意正负号，因为符号错误是常见的失分点。
∫ 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
For tangent and cotangent, you can rewrite them as sine over cosine or cosine over sine and integrate using substitution. For instance, ∫ tan x dx = ∫ (sin x / cos x) dx = −ln|cos x| + C. Although the integrals of sec x and csc x are more involved, it is beneficial to be familiar with them.
对于正切和余切，可以将其重写为正弦比余弦或余弦比正弦，然后利用代换积分。例如，∫ tan x dx = ∫ (sin x / cos x) dx = −ln|cos x| + C。尽管 sec x 和 csc x 的积分较为复杂，但熟悉它们是有益的。
∫ tan x dx = −ln|cos x| + C = ln|sec x| + C
∫ cot x dx = ln|sin x| + C
5. The Method of Substitution (u-Substitution) | 换元积分法（u-代换）
Substitution is the most powerful technique for integrating composite functions — functions that are the result of a chain of functions. The idea is to choose an inner function u = g(x) so that the integral contains its derivative du = g'(x) dx. This often simplifies a complicated integral into a standard form.
代换法是积分复合函数——即函数嵌套结果——的最有力技巧。其思想是选择一个内层函数 u = g(x)，使得被积式中包含其导数 du = g'(x) dx。这往往能将一个复杂的积分简化为标准形式。
∫ f(g(x)) g'(x) dx = ∫ f(u) du
For a definite integral, you must change the limits of integration to match the new variable u, or revert to x after integrating. Consider ∫ 2x·sin(x²) dx. Let u = x², then du = 2x dx. The integral becomes ∫ sin u du = −cos u + C = −cos(x²) + C. Another common pattern is ∫ (ln x)/x dx: let u = ln x, du = (1/x) dx, yielding ∫ u du = (1/2)(ln x)² + C.
对于定积分，必须将积分上下限也转换为新变量 u 的值，或者在积分后换回 x。考虑 ∫ 2x·sin(x²) dx。令 u = x²，则 du = 2x dx。积分变为 ∫ sin u du = −cos u + C = −cos(x²) + C。另一个常见模式是 ∫ (ln x)/x dx：令 u = ln x，du = (1/x) dx，得到 ∫ u du = (1/2)(ln x)² + C。
Successful substitution hinges on identifying a part of the integrand whose derivative also appears. Practice helps you recognise these patterns quickly.
代换成功的关键在于识别出被积函数中某一部分的导数也同时出现。勤加练习有助于快速识别这些模式。
6. Integration by Parts | 分部积分法
Integration by parts is the integral counterpart of the product rule for differentiation. It is used primarily when the integrand is a product of two unlike functions, such as a polynomial multiplied by an exponential, logarithm, or trigonometric function. The formula is derived from the product rule and rearranged.
分部积分法是微分乘法法则的积分对应。它主要用于被积函数是两个不同类型函数的乘积时，例如多项式乘以指数、对数或三角函数。该公式由乘法法则推导并重新排列而得。
∫ u dv = uv − ∫ v du
Choosing u and dv wisely is crucial. The LIATE rule provides a helpful priority order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Choose u from the category that appears first in this list, and let dv be the rest of the expression including dx. For example, in ∫ x eˣ dx, choose u = x (Algebraic) and dv = eˣ dx. Then du = dx and v = eˣ, giving ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.
合理选择 u 和 dv 至关重要。LIATE 法则提供了一个有用的优先级顺序：对数函数、反三角函数、代数函数、三角函数、指数函数。从该列表中排在最前的类别中选取 u，并令 dv 为包含 dx 的剩余部分。例如，在 ∫ x eˣ dx 中，选择 u = x（代数），dv = eˣ dx。则 du = dx，v = eˣ，得到 ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。
For ∫ ln x dx, there is no apparent product, but we treat it as u = ln x and dv = dx. This yields the standard result x ln x − x + C. In some cases, you must apply integration by parts twice and solve for the original integral algebraically, such as when integrating eˣ sin x.
对于 ∫ ln x dx，表面上没有乘积，但我们将其视为 u = ln x 和 dv = dx。这便得到标准结果 x ln x − x + C。在某些情形下，需要两次使用分部积分并代数求解原积分，例如积分 eˣ sin x 时。
7. Integration Using Partial Fractions | 部分分式积分法
Partial fractions is a technique for integrating rational functions, which are ratios of polynomials. If the degree of the numerator is greater than or equal to that of the denominator, perform polynomial long division first. Then factor the denominator and express the fraction as a sum of simpler fractions whose denominators are those factors.
部分分式是一种积分有理函数（即多项式之比）的技巧。如果分子的次数大于或等于分母的次数，需先进行多项式长除法。然后对分母因式分解，并将该分式表示为一系列分母为这些因子的更简单分式之和。
For a denominator with distinct linear factors, e.g., x²−x−2 = (x−2)(x+1), write 1/(x²−x−2) = A/(x−2) + B/(x+1). Solve for constants A and B by equating numerators. Then integrate each term: ∫ A/(x−2) dx + ∫ B/(x+1) dx = A ln|x−2| + B ln|x+1| + C.
对于分母为不同线性因子的情况，例如 x²−x−2 = (x−2)(x+1)，将其写成 1/(x²−x−2) = A/(x−2) + B/(x+1)。通过令分子相等解出常数 A 和 B。然后逐项积分：∫ A/(x−2) dx + ∫ B/(x+1) dx = A ln|x−2| + B ln|x+1| + C。
∫ (2x+3)/(x²−1) dx = ∫ (5/2)/(x−1) dx + ∫ (−1/2)/(x+1) dx
Repeated linear factors or irreducible quadratic factors require adjusted forms, but the core idea remains the same: decompose the complex fraction into a sum of simpler ones that you can integrate using basic logarithm or arctangent rules.
重复线性因子或不可约二次因子需要调整形式，但核心思想不变：将复杂分式分解为可分别利用基本对数或反正切规则积分的更简单分式之和。
8. Definite Integrals and the Fundamental Theorem of Calculus | 定积分与微积分基本定理
The Fundamental Theorem of Calculus (FTC) bridges the concept of antiderivatives with the area under a curve. If F is any antiderivative of f on an interval [a, b], the definite integral of f from a to b equals the net change in F over that interval. This powerful theorem allows you to evaluate definite integrals without going back to the limit of Riemann sums.
微积分基本定理 (FTC) 将反导数的概念与曲线下的面积联系了起来。如果 F 是 f 在区间 [a, b] 上的任意一个反导数，那么 f 从 a 到 b 的定积分等于 F 在该区间上的净变化。这一定理使你无需回到黎曼和的极限即可计算定积分。
∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x)
For example, to find the area under f(x) = x² from x = 0 to x = 2, an antiderivative is F(x) = (1/3)x³. Then ∫₀² x² dx = (1/3)(2)³ − (1/3)(0)³ = 8/3. Notice that the constant C cancels out, so it is omitted in definite integral evaluations.
例如，要计算 f(x) = x² 从 x = 0 到 x = 2 下的面积，一个反导数是 F(x) = (1/3)x³。那么 ∫₀² x² dx = (1/3)(2)³ − (1/3)(0)³ = 8/3。注意常数 C 会抵消，因此在计算定积分时通常省略。
The FTC also connects differentiation and integration as inverse processes, solidifying the unity of calculus. Always evaluate antiderivatives at the upper limit first, then subtract the value at the lower limit.
微积分基本定理还将微分与积分连接为互逆过程，巩固了微积分的统一性。始终先计算上界处的反导数值，再减去下界处的值。
9. Properties of Definite Integrals | 定积分的性质
Definite integrals possess several algebraic properties that simplify calculations and provide deeper insight. Reversing the limits of integration changes the sign of the integral. This property is extremely useful when you need to flip bounds or correct an orientation.
定积分具有若干代数性质，能够简化计算并提供更深层次的理解。颠倒积分上下限会改变积分的符号。这一性质在需要翻转界限或修正方向时极为有用。
∫ₐᵇ f(x) dx = −∫ₒᵃ f(x) dx
The integral over an interval of zero width is always zero: ∫ₐᵃ f(x) dx = 0. Additionally, the interval additivity property states that integrating over [a, c] is the same as integrating over [a, b] plus [b, c], provided the function is integrable on these subintervals.
零宽区间上的积分永远为零：∫ₐᵃ f(x) dx = 0。此外，区间可加性指出，在 [a, c] 上积分等同于分别在 [a, b] 和 [b, c] 上积分再求和，只要函数在这些子区间上可积。
∫ₐᶜ f(x) dx = ∫ₐᵇ f(x) dx + ∫ₒᶜ f(x) dxPublished by TutorHao | IB Mathematics Revision Series | aleveler.com
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