IB Maths: Integration Key Points | IB 数学：积分 考点精讲

📚 IB Maths: Integration Key Points | IB 数学：积分 考点精讲

Integration is a cornerstone of calculus in the IB Mathematics curriculum, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI). It extends differentiation and serves as a tool for calculating areas, volumes, solving differential equations and modelling real-world phenomena. A solid grasp of integration techniques and their applications is essential for success in IB exams. This guide covers key integration concepts, standard integrals, methods like substitution and integration by parts, definite integrals, area, volume of revolution, kinematics, numerical integration, probability density functions and improper integrals—tightly aligned with the IB syllabus.

积分是IB数学课程中微积分的基石，在分析与方法(AA)和应用与解释(AI)两门课程中都占据核心地位。它不仅是微分的逆运算，更是计算面积、体积、求解微分方程以及建立现实世界模型的有力工具。透彻掌握积分技巧及其应用对于IB考试的成功至关重要。本指南涵盖核心积分概念、基本积分公式、换元积分法与分部积分法等技巧、定积分、面积、旋转体体积、运动学、数值积分、概率密度函数以及反常积分，完全贴合IB考纲要求。

1. Indefinite Integrals and Standard Forms | 不定积分与基本积分公式

An indefinite integral, also called an antiderivative, reverses differentiation. If F'(x) = f(x), then ∫ f(x) dx = F(x) + C, where C is an arbitrary constant. This constant is essential because differentiation wipes out any constant term. Memorising the standard integrals below is fundamental to IB integration questions.

不定积分也称为反导数，是微分的逆运算。如果 F'(x) = f(x)，那么 ∫ f(x) dx = F(x) + C，其中 C 为任意积分常数。常数项必不可少，因为微分会消除所有常数项。熟记以下基本积分公式是解决IB积分问题的基础。

• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
• ∫ 1/x dx = ln|x| + C
• ∫ eˣ dx = eˣ + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C
• ∫ 1/(1+x²) dx = arctan x + C
• ∫ 1/√(1-x²) dx = arcsin x + C

• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
• ∫ 1/x dx = ln|x| + C
• ∫ eˣ dx = eˣ + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C
• ∫ 1/(1+x²) dx = arctan x + C
• ∫ 1/√(1-x²) dx = arcsin x + C

2. Linearity of Integration | 积分的线性性质

Integration is a linear operator. This means ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx, where a and b are constants. You can integrate term by term, pulling constant coefficients outside. This property allows us to handle polynomials and linear combinations of standard functions efficiently.

积分是线性算子，满足 ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx，其中 a、b 为常数。我们可以逐项积分，并将常数系数提到积分号外。这一性质使我们能够高效地处理多项式以及标准函数的线性组合。

Example: ∫ (3x² + 2x – 5) dx = 3·x³/3 + 2·x²/2 – 5x + C = x³ + x² – 5x + C. Always check by differentiation.

例如：∫ (3x² + 2x – 5) dx = 3·x³/3 + 2·x²/2 – 5x + C = x³ + x² – 5x + C。记得通过求导验证结果。

3. Integration by Substitution | 换元积分法

The substitution method reverses the chain rule. If an integrand can be written as f(g(x))·g'(x), set u = g(x) so that du = g'(x) dx. The integral becomes ∫ f(u) du, which is often simpler. For definite integrals, remember to change the limits.

换元积分法是链式法则的逆用。若被积函数可写成 f(g(x))·g'(x) 的形式，令 u = g(x)，则 du = g'(x) dx。积分转化为 ∫ f(u) du，通常更易求解。在处理定积分时，务必记得同步替换积分上下限。

Example: ∫ 2x cos(x²) dx. Let u = x², then du = 2x dx. The integral becomes ∫ cos u du = sin u + C = sin(x²) + C. For ∫₀² 2x cos(x²) dx, new limits: when x=0, u=0; when x=2, u=4, so the value is sin 4 – sin 0 = sin 4.

例如：∫ 2x cos(x²) dx。设 u = x²，则 du = 2x dx。积分化为 ∫ cos u du = sin u + C = sin(x²) + C。若求 ∫₀² 2x cos(x²) dx，换限：x=0 时 u=0；x=2 时 u=4，因此定积分为 sin 4 – sin 0 = sin 4。

4. Integration by Parts (HL) | 分部积分法（仅HL）

The integration by parts formula is derived from the product rule: ∫ u dv = uv – ∫ v du. Success depends on choosing u and dv wisely so that ∫ v du is simpler than the original integral. A common mnemonic is LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to prioritise u.

分部积分法源自乘积求导法则：∫ u dv = uv – ∫ v du。成功的关键在于合理选择 u 和 dv，使得 ∫ v du 比原积分更易求解。常见的优先顺序为 LIATE（对数、反三角、代数、三角、指数），用于确定 u。

Example 1: ∫ x eˣ dx. Let u = x, dv = eˣ dx; then du = dx, v = eˣ. ∫ x eˣ dx = x eˣ – ∫ eˣ dx = x eˣ – eˣ + C. Example 2: ∫ ln x dx. Take u = ln x, dv = dx; then du

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