📚 IB Mathematics: Core Concepts of Integration | IB数学：积分核心知识点梳理
Integration is one of the most powerful tools in IB Mathematics, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. It allows us to reverse differentiation, calculate areas, volumes, and accumulated change, and solve differential equations that model real-world phenomena. This article organises the essential integration concepts you need for your IB exams, with clear formulas and techniques.
积分是IB数学中最强大的工具之一，出现在分析与方法（AA）和应用与解释（AI）课程中。它使我们能够逆推微分、计算面积、体积和累积变化，并求解模拟现实世界现象的微分方程。本文围绕IB考试所需的核心积分知识点进行梳理，提供清晰的公式与技巧。
1. Indefinite Integrals and Basic Rules | 不定积分与基本法则
An indefinite integral, also called an antiderivative, is a function F such that F'(x) = f(x). It is written as ∫ f(x) dx = F(x) + C, where C is the constant of integration. Since the derivative of any constant is zero, the constant must always be included.
不定积分也称为原函数，是满足 F'(x) = f(x) 的函数 F。记作 ∫ f(x) dx = F(x) + C，其中 C 为积分常数。因为任意常数的导数都是零，所以积分常数必须保留。
The power rule is the most frequently used rule: if n ≠ −1, then
幂法则是使用频率最高的规则：若 n ≠ −1，则
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
For n = −1, the rule changes because division by zero is impossible: ∫ x⁻¹ dx = ln|x| + C. More generally, ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n+1)] + C for n ≠ −1, and ∫ (ax + b)⁻¹ dx = (1/a) ln|ax + b| + C.
当 n = −1 时，由于不能除以零，法则变为：∫ x⁻¹ dx = ln|x| + C。更一般地，∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n+1)] + C（n ≠ −1），以及 ∫ (ax + b)⁻¹ dx = (1/a) ln|ax + b| + C。
Integration is linear: ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx. This means you can integrate term by term.
积分具有线性：∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx。这意味着你可以逐项积分。
2. Integration by Substitution | 换元积分法
Substitution is the reverse of the chain rule. It is used when the integrand contains a composite function and the derivative of its inner function as a factor. The general idea is to set u = g(x), then du = g'(x) dx, and rewrite the entire integral in terms of u.
换元法是链式法则的逆运算。当被积函数含有复合函数，并且包含其内层函数的导数因子时，可以使用换元。一般思路是令 u = g(x)，则 du = g'(x) dx，然后把整个积分改写为关于 u 的积分。
For example, ∫ 2x cos(x²) dx. Let u = x², so du = 2x dx. The integral becomes ∫ cos u du = sin u + C = sin(x²) + C.
例如，∫ 2x cos(x²) dx。令 u = x²，则 du = 2x dx。积分变为 ∫ cos u du = sin u + C = sin(x²) + C。
For definite integrals, you must also change the limits of integration. If x runs from a to b, then u runs from g(a) to g(b). Alternatively, you can convert back to x before applying the original limits. Always check whether your answer is in the correct variable.
对于定积分，必须同时更换积分上下限。若 x 从 a 到 b，则 u 从 g(a) 到 g(b)。另一种方法是先换回 x 再代入原上下限。始终确认答案的变量正确。
3. Integration by Parts | 分部积分法
Integration by parts is the reverse of the product rule. It is useful for products of functions such as x eˣ, x sin x, or ln x. The formula is:
分部积分法是乘积法则的逆运算，适用于诸如 x eˣ、x sin x 或 ln x 等函数乘积。公式为：
∫ u dv = uv − ∫ v du
To apply it, choose u to be a function that becomes simpler when differentiated (often a polynomial), and choose dv to be the remaining part that can be integrated easily. For example, ∫ x eˣ dx: let u = x, dv = eˣ dx, so du = dx, v = eˣ. Then ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.
应用时，选择 u 为求导后变得更简单的函数（通常是多项式），选择 dv 为容易积分的剩余部分。例如，∫ x eˣ dx：令 u = x，dv = eˣ dx，则 du = dx，v = eˣ。于是 ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。
Sometimes you need to apply parts twice, especially when both factors are trigonometric or exponential. In IB AA HL, the LIATE rule (Logarithms, Inverse trig, Algebraic, Trigonometric, Exponential) can help you choose u.
有时需要连续使用两次分部积分，尤其是当两个因子都是三角或指数函数时。在IB AA HL中，LIATE规则（对数、反三角、代数、三角、指数）可帮助你选择 u。
4. The Fundamental Theorem of Calculus | 微积分基本定理
The Fundamental Theorem of Calculus connects differentiation and integration. It states that if f is continuous on [a, b] and F is any antiderivative of f, then
微积分基本定理将微分与积分联系起来。它指出：若 f 在 [a, b] 上连续，F 是 f 的任意一个原函数，则
∫ₐᵇ f(x) dx = F(b) − F(a)
This theorem allows you to evaluate definite integrals exactly, without computing the limit of Riemann sums. It also states that the derivative of an integral with a variable upper limit is the original function: d/dx [∫ₐˣ f(t) dt] = f(x).
这一定理使你可以精确计算定积分，而不必计算黎曼和的极限。它同时指出：变上限积分的导数等于被积函数本身，即 d/dx [∫ₐˣ f(t) dt] = f(x)。
In IB problems, you may be asked to find the derivative of an integral using this principle, or to evaluate a definite integral by first finding an antiderivative. Always apply the limits after integrating, and be careful with signs when the upper limit is a function of x.
在IB题目中，你可能需要利用该原理求积分的导数，或者通过先找原函数来计算定积分。积分后务必代入上下限；当上限为 x 的函数时，需注意链式法则和符号。
5. Definite Integrals and Area | 定积分与面积
A definite integral ∫ₐᵇ f(x) dx represents the signed area between the curve y = f(x) and the x-axis, from x = a to x = b. Area above the x-axis contributes positive value; area below contributes negative value.
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的有向面积。x 轴上方的面积贡献正值，下方的面积贡献负值。
To find the total geometric area bounded by a curve and the x-axis, split the interval at the zeros of f and integrate the absolute value: ∫ₐᵇ |f(x)| dx. For the area between two curves f(x) and g(x), use ∫ₐᵇ |f(x) − g(x)| dx, where a and b are their intersection points or given limits.
要求曲线与 x 轴围成的几何总面积，需要在 f 的零点处拆分区间并积分绝对值：∫ₐᵇ |f(x)| dx。对于两条曲线 f(x) 与 g(x) 之间的面积，使用 ∫ₐᵇ |f(x) − g(x)| dx，其中 a、b 为交点或给定端点。
Remember to sketch the curves when necessary. If the functions cross within the interval, you must split the integral at the intersection point, because the upper/lower functions change.
必要时先画出曲线草图。如果函数在区间内相交，必须在交点处拆开积分，因为上下函数关系发生了改变。
6. Integrals of Common Functions | 常用函数积分
You should memorise the following standard integrals, which appear frequently in IB exams:
以下标准积分在IB考试中频繁出现，需要牢记：
| Function | Integral |
| ∫ eˣ dx | eˣ + C |
| ∫ sin x dx | −cos x + C |
| ∫ cos x dx | sin x + C |
| ∫ sec² x dx | tan x + C |
| ∫ 1/x dx | ln|x| + C |
| ∫ 1/(1+x²) dx | arctan x + C |
| ∫ 1/√1−x² dx | arcsin x + C |
For rational functions, use algebraic manipulation or substitution. For example, ∫ 1/(x² + a²) dx = (1/a) arctan(x/a) + C, which can be derived using the substitution x = a tan u.
对于有理函数，可使用代数变形或换元。例如，∫ 1/(x² + a²) dx = (1/a) arctan(x/a) + C，可通过令 x = a tan u 推导得出。
7. Integration by Partial Fractions | 部分分式积分法
Partial fractions are used to integrate rational functions of the form P(x)/Q(x), where the denominator can be factored into linear or quadratic factors. The idea is to decompose the fraction into simpler fractions that can be integrated term by term.
部分分式用于积分形如 P(x)/Q(x) 的有理函数，其中分母可以分解为一次或二次因式。其思想是把原分式拆成更简单、可逐项积分的分式之和。
For example, to integrate 1/[(x−1)(x−2)], write 1/[(x−1)(x−2)] = A/(x−1) + B/(x−2). Solving gives A = −1, B = 1, so the integral becomes −ln|x−1| + ln|x−2| + C = ln|(x−2)/(x−1)| + C.
例如，积分 1/[(x−1)(x−2)]，设 1/[(x−1)(x−2)] = A/(x−1) + B/(x−2)。解得 A = −1，B = 1，因此积分为 −ln|x−1| + ln|x−2| + C = ln|(x−2)/(x−1)| + C。
In IB AA HL, you may also need to use partial fractions when the denominator has repeated linear factors or irreducible quadratic factors. For repeated factors like (x−a)², the decomposition includes both A/(x−a) and B/(x−a)².
在IB AA HL中，还可能遇到分母含重因式或不可约二次因式的情况。对于 (x−a)² 这类重因式，分解式中需包含 A/(x−a) 和 B/(x−a)²。
8. Differential Equations and Separation of Variables | 微分方程与变量分离法
A differential equation is an equation involving a derivative. The simplest type in IB is a first-order separable differential equation, which can be written in the form dy/dx = f(x) g(y). The method of separation of variables moves all y-terms to one side and all x-terms to the other:
微分方程是含有导数的方程。IB中最简单类型是一阶可分离变量微分方程，可写成 dy/dx = f(x) g(y) 的形式。变量分离法将含 y 的项移到一边，含 x 的项移到另一边：
∫ 1/g(y) dy = ∫ f(x) dx
After integrating both sides, you obtain an implicit relation between x and y. If initial conditions are given, use them to determine the constant C. For example, if dy/dx = 2xy, then ∫ 1/y dy = ∫ 2x dx, giving ln|y| = x² + C, hence y = A ex².
两边积分后，得到 x 与 y 的隐式关系。若给定初始条件，则用于确定常数 C。例如，若 dy/dx = 2xy，则 ∫ 1/y dy = ∫ 2x dx，得到 ln|y| = x² + C，因此 y = A ex²。
Always state the domain of the solution and check for singular solutions when dividing by a function that could be zero, such as g(y) = 0.
注意写出解的定义域，并检查除以可能为零的函数（如 g(y)=0）时是否存在奇解。
9. Volumes of Revolution | 旋转体体积
When a region under a curve y = f(x) from x = a to x = b is rotated about the x-axis, the volume of the resulting solid is given by the disk method:
当曲线 y = f(x) 下方区域从 x = a 到 x = b 绕 x 轴旋转时，所得立体体积由圆盘法给出：
V = π ∫ₐᵇ [f(x)]² dx
If the region is rotated about the y-axis, write x as a function of y, say x = g(y), with y from c to d, then V = π ∫_cd [g(y)]² dy. In some IB AI and AA problems, you may be asked to find the volume between two curves; then use V = π ∫ₐᵇ ([f(x)]² − [g(x)]²) dx, where f is the outer radius and g is the inner radius.
若区域绕 y 轴旋转，则将 x 表示为 y 的函数，如 x = g(y)，y 从 c 到 d，则 V = π ∫_cd [g(y)]² dy。在一些IB AI和AA题目中，可能要求两曲线间旋转体体积，此时使用 V = π ∫ₐᵇ ([f(x)]² − [g(x)]²) dx，其中 f 为外半径，g 为内半径。
Always draw the region and identify the axis of rotation clearly. If the axis is not x = 0 or y = 0, the integrand must be adjusted using the distance from the curve to the axis.
务必画出区域并明确旋转轴。若旋转轴不是 x=0 或 y=0，则需要用曲线到轴的距离来调整被积函数。
10. Kinematics and Accumulated Change | 运动学与累积变化
Integration is essential in kinematics. If velocity v(t) is the derivative of displacement s(t), then displacement is the integral of velocity: s(t) = ∫ v(t) dt. The distance travelled over [a, b] is ∫ₐᵇ |v(t)| dt, while the displacement is ∫ₐᵇ v(t) dt.
积分在运动学中至关重要。若速度 v(t) 是位移 s(t) 的导数，则位移是速度的积分：s(t) = ∫ v(t) dt。在 [a, b] 上经过的路程为 ∫ₐᵇ |v(t)| dt，而位移为 ∫ₐᵇ v(t) dt。
Acceleration is the derivative of velocity, so velocity can be recovered from acceleration by integration: v(t) = ∫ a(t) dt. Initial conditions are necessary to determine the constant of integration.
加速度是速度的导数，因此通过对加速度积分可以恢复速度：v(t) = ∫ a(t) dt。必须利用初始条件确定积分常数。
More generally, integration measures total change from a rate of change. If a quantity changes at rate R(t), then the total change from time t₁ to t₂ is ∫t₁t₂ R(t) dt. This concept appears in population growth, water flow, economics, and many IB AI applications.
更一般地，积分通过变化率来度量总变化量。若某量以速率 R(t) 变化，则在时间 t₁ 到 t₂ 内的总变化为 ∫t₁t₂ R(t) dt。这一概念出现在人口增长、水流、经济学以及许多IB AI应用题中。
11. Improper Integrals and Convergence | 反常积分与收敛性
An improper integral occurs when the interval is infinite or the integrand becomes unbounded near an endpoint. For example, ∫₁∞ 1/x² dx is improper because the upper limit is infinite. It is evaluated as a limit:
当积分区间无穷或被积函数在端点附近无界时，即出现反常积分。例如，∫₁∞ 1/x² dx 因为上限无穷而为反常积分。它通过极限来计算：
∫ₐ^∞ f(x) dx = limt→∞ ∫ₐᵗ f(x) dx
If the limit exists and is finite, the improper integral converges; otherwise it diverges. A typical example is ∫₁∞ 1/x² dx = 1, while ∫₁∞ 1/x dx diverges to infinity.
若极限存在且有限，则反常积分收敛；否则发散。典型例子是 ∫₁∞ 1/x² dx = 1，而 ∫₁∞ 1/x dx 发散至无穷。
For integrands with vertical asymptotes, split the integral at the singular point and take one-sided limits. In IB AA HL, you may compare an improper integral with a known convergent or divergent p-integral to determine convergence.
对于含垂直渐近线的被积函数，需在奇点处拆分积分并取单侧极限。在IB AA HL中，可将反常积分与已知收敛或发散的 p-积分进行比较，以判断收敛性。
12. Key Exam Advice | 考试关键建议
Master the table of standard integrals and the three core techniques: substitution, parts, and partial fractions. In definite integrals, never forget to change limits when substituting, or to return to the original variable before applying the original limits.
掌握标准积分表以及三大核心技巧：换元、分部、部分分式。在定积分中，换元时不要忘记更换上下限，或者先换回原变量再代入原上下限。
When computing areas, use absolute values or split the region at intersections. For volumes, clearly identify the axis and the correct radius function. For differential equations, always apply initial conditions to solve for C and check the validity of the solution.
计算面积时使用绝对值或在交点处拆分区域。求体积时，明确旋转轴和正确的半径函数。解微分方程时，务必用初始条件求出 C，并检验解的有效性。
Finally, practice reading questions carefully: IB problems often combine integration with other topics such as functions, graphs, or modelling. Write out every step clearly, as method marks are often awarded.
最后，仔细审题：IB题目常把积分与函数、图像或建模等其他主题结合。每一步都要书写清楚，因为步骤分通常很关键。
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