📚 Integral Calculus Key Points Revision | IB WJEC 数学积分考点精讲
Integration is one of the two central pillars of calculus, alongside differentiation. In both the IB Diploma and WJEC A-level Mathematics specifications, integral calculus appears across pure and applied topics—from reversing derivatives to calculating areas under curves, volumes of revolution, and solving real-world motion problems. A deep grasp of integration techniques and their geometric meaning is essential for top-band marks. This revision guide distils the key concepts, standard results, and problem-solving strategies you need to master integration for your exams.
积分是微积分的两大支柱之一，与微分并列。在 IB 文凭和 WJEC A-level 数学课程中，积分贯穿纯数与应用的各个主题——从逆求导函数到计算曲线下的面积、旋转体体积以及求解现实中的运动问题。深刻理解积分技巧及其几何意义是获取高分的关键。这篇考点精讲提炼了你必须掌握的核心概念、标准结果和解题策略。
We will cover indefinite and definite integrals, the power rule, integrals of exponential, logarithmic and trigonometric functions, substitution, integration by parts, partial fractions, area and volume applications, kinematics, and elementary differential equations—all tightly aligned with the IB Analysis & Approaches and WJEC pure and applied modules.
我们将涵盖不定积分与定积分、幂函数法则、指数函数、对数函数和三角函数的积分、换元积分法、分部积分法、部分分式、面积与体积应用、运动学以及简单的微分方程——全部紧扣 IB Analysis & Approaches 和 WJEC 纯数及应用模块。
1. Indefinite Integrals and the Constant of Integration | 不定积分与积分常数
An indefinite integral of a function f(x) is written as ∫ f(x) dx and represents the family of all antiderivatives of f. Since the derivative of any constant is zero, we must add a constant of integration, C, to account for all possible vertical shifts of the original function. In IB and WJEC exams, forgetting the +C in an indefinite integral will cost you at least one mark, unless the question specifically asks for a particular antiderivative.
函数 f(x) 的不定积分记作 ∫ f(x) dx，它表示 f 的所有原函数构成的族。因为任意常数的导数都是零，所以必须加上积分常数 C，以涵盖原函数所有可能的竖直平移。在 IB 和 WJEC 考试中，如果不定积分漏写 +C，除非题目明确要求某个特定的原函数，否则至少会扣掉一分。
The general notation is: if F'(x) = f(x), then ∫ f(x) dx = F(x) + C. This close relationship with differentiation is the foundation of all integration techniques. Always check your integration result by differentiating mentally; if you obtain the original integrand, your answer is correct.
一般记法为：若 F'(x) = f(x)，则 ∫ f(x) dx = F(x) + C。这种与微分的紧密关系是所有积分技巧的基础。积分后最好在心里求导检验一下：如果能还原出被积函数，答案就是正确的。
In the IB, questions often start with ‘Find ∫ (2x³ – 5x + 1) dx’ and expect both the integrated expression and the +C. WJEC papers likewise emphasize notation and will penalise omitted constants in indefinite integrals.
IB 试题中常出现类似“求 ∫ (2x³ – 5x + 1) dx”的题目，要求写出积分后的表达式并加上 +C。WJEC 试卷同样强调符号规范性，不定积分中漏掉常数会被扣分。
2. The Power Rule and Polynomial Integrals | 幂函数法则与多项式积分
The most fundamental integration rule is the reverse of the power rule for differentiation. For any real number n ≠ –1, ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. This formula can be extended to functions of the form (ax+b)ⁿ through substitution, but the basic principle remains: increase the exponent by one and divide by the new exponent.
最根本的积分法则是幂函数求导的逆运算。对于任意实数 n ≠ –1，∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。该公式可通过换元推广到 (ax+b)ⁿ 等形式，但基本原则不变：指数加一，然后除以新的指数。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ –1
When integrating a polynomial, simply integrate term by term. For example, ∫ (4x³ – 2x + 7) dx = x⁴ – x² + 7x + C. Remember to apply the rule to each term independently and to include a single constant C for the whole expression.
对多项式积分时，只需逐项积分即可。例如 ∫ (4x³ – 2x + 7) dx = x⁴ – x² + 7x + C。要记得对每一项独立使用幂函数法则，并在整个表达式末尾加上一个常数 C。
The case n = –1 must be treated separately: ∫ x⁻¹ dx = ∫ (1/x) dx = ln|x| + C. This is a common source of error when students blindly apply the power rule and obtain an undefined division. Always watch out for rational functions where the numerator is the derivative of the denominator; these integrate directly to a natural logarithm in WJEC and IB syllabuses.
当 n = –1 时必须单独处理：∫ x⁻¹ dx = ∫ (1/x) dx = ln|x| + C。如果学生盲目套用幂函数法则，就会得到除数为零的错误，这是一个常见失误。当有理函数的分子恰好是分母的导数时，可直接积出自然对数，这在 WJEC 和 IB 大纲中都很常见。
3. Integrating Exponential and Logarithmic Functions | 指数函数与对数函数的积分
Exponential functions are exceptionally simple to integrate. The integral of eˣ is itself: ∫ eˣ dx = eˣ + C. For the more general exponential aˣ, rewrite as e^(x ln a) or use the formula ∫ aˣ dx = aˣ / ln a + C. IB and WJEC focus mainly on the base e, but knowing the general result is useful for applications.
指数函数的积分极其简单。eˣ 的积分就是它本身：∫ eˣ dx = eˣ + C。对于更一般的指数 aˣ，可将其化为 e^(x ln a) 或直接使用公式 ∫ aˣ dx = aˣ / ln a + C。IB 和 WJEC 主要考察以 e 为底的指数函数，但了解一般形式对应用很有帮助。
When the exponent is a linear function, like e^(kx), the integral becomes (1/k) e^(kx) + C. This is a special case of the reverse chain rule and can be verified by differentiating. In WJEC applied modules, you will frequently integrate expressions like e^(–kt) when modelling decay.
当指数为线性函数时，例如 e^(kx)，其积分为 (1/k) e^(kx) + C，这是逆链式法则的特殊情况，可通过求导来验证。在 WJEC 应用模块中，描述衰变时常需要积分形如 e^(–kt) 的表达式。
For logarithms, there is no direct simple integral of ln x, but integration by parts gives ∫ ln x dx = x ln x – x + C. However, the integral ∫ (1/x) dx = ln|x| + C is a standard result you must know instantly. The absolute value is important: the domain of the logarithm requires a positive argument, and the |x| notation ensures correctness for negative x.
对数函数本身没有直接简单的积分，但利用分部积分法可得 ∫ ln x dx = x ln x – x + C。然而 ∫ (1/x) dx = ln|x| + C 是一个必须迅速反应的标准结果。注意绝对值：对数函数的定义域要求参数为正，|x| 的记法保证了在 x 为负时结果也正确。
4. Integration of Trigonometric Functions | 三角函数的积分
Trigonometric integrals appear regularly in both IB and WJEC pure mathematics. The basic integrals to memorise are:
三角函数积分在 IB 和 WJEC 纯数学中定期出现。需要熟记的基本积分有：
∫ sin x dx = –cos x + C
∫ cos x dx = sin x + C
∫ sec² x dx = tan x + C
More advanced integrals, such as ∫ tan x dx = ln|sec x| + C or ∫ cot x dx = ln|sin x| + C, are derived using substitution and may appear in higher-tier IB HL questions. WJEC pure papers typically provide such integrals if needed, but it is wise to be familiar with the derivations.
更进阶的积分，如 ∫ tan x dx = ln|sec x| + C 或 ∫ cot x dx = ln|sin x| + C，可借助换元法推导，并可能出现在 IB HL 高阶题中。WJEC 纯数试卷通常会在需要时给出这类积分，但熟悉推导过程更为稳妥。
When the argument is a multiple of x, e.g. sin(kx), the integral becomes –(1/k) cos(kx) + C. This follows from the same reverse chain rule idea as for exponential functions. Always differentiate your answer to confirm the coefficient is correct.
当参数是 x 的倍数，比如 sin(kx) 时，积分为 –(1/k) cos(kx) + C。这与指数函数中的逆链式法则思路一致。务必通过求导来验证系数是否正确。
Squaring trigonometric identities: integrals of sin²x or cos²x appear frequently. Use the power-reducing identities cos 2x = 1 – 2 sin²x or cos 2x = 2 cos²x – 1 to rewrite the integrand before integrating. For example, ∫ sin²x dx = ∫ (1 – cos 2x)/2 dx = x/2 – (sin 2x)/4 + C.
三角函数的平方积分也很常见：利用降幂公式 cos 2x = 1 – 2 sin²x 或 cos 2x = 2 cos²x – 1 将被积函数改写后再积分。例如，∫ sin²x dx = ∫ (1 – cos 2x)/2 dx = x/2 – (sin 2x)/4 + C。
5. Integration by Substitution | 换元积分法
Substitution, or the reverse chain rule, is used when an integrand contains a function and its derivative. The idea is to let u = g(x), so that du = g'(x) dx, transforming ∫ f(g(x)) g'(x) dx into ∫ f(u) du, which is hopefully simpler. This method is central to both IB analysis and WJEC pure modules.
换元积分法（逆链式法则）适用于被积函数包含一个函数及其导数的情况。其核心思路是令 u = g(x)，则 du = g'(x) dx，从而把 ∫ f(g(x)) g'(x) dx 转化为 ∫ f(u) du，后者往往更容易计算。该方法在 IB 分析与 WJEC 纯数模块中都处于核心地位。
∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x)
It is crucial to convert all parts of the integral, including the dx, and to revert to the original variable after integration unless computing a definite integral, in which case you can change the limits. In IB exams, clearly showing the substitution and the working for changing limits is required for full marks.
必须将被积函数的所有部分，包括 dx，都转换为新变量，并在积分后回代为原变量——除非是计算定积分，此时可以同时转换积分上下限。在 IB 考试中，清晰展示换元过程和积分限的变换步骤才能拿到满分。
Examples often include integrands with a linear inner function raised to a power, or exponential and trigonometric composites. For instance, ∫ 2x e^(x²) dx: let u = x², then du = 2x dx, so the integral becomes ∫ eᵘ du = eᵘ + C = e^(x²) + C.
常见的例子有：线性内层函数的幂次、指数与三角复合函数。例如 ∫ 2x e^(x²) dx：令 u = x²，则 du = 2x dx，积分变为 ∫ eᵘ du = eᵘ + C = e^(x²) + C。
6. Integration by Parts | 分部积分法
Integration by parts is the reverse of the product rule for differentiation and is required in IB HL and optional in some WJEC further units. The formula is:
分部积分法是函数乘积求导法则的逆运算，在 IB HL 中是必考内容，在部分 WJEC 进阶单元中也会涉及。公式为：
∫ u dv = uv – ∫ v du
To apply this technique, you must split the integrand into u and dv such that dv is easy to integrate and u simplifies when differentiated. A common mnemonic is LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to help choose u.
应用该方法时，需要将被积函数拆分为 u 和 dv，使得 dv 容易积分，而 u 在求导后会变得更简单。常用的选择顺序助记法是 LIATE（对数、反三角、代数、三角、指数），用于确定 u。
A classical example is ∫ x eˣ dx: let u = x, dv = eˣ dx, then du = dx, v = eˣ, giving xeˣ – ∫ eˣ dx = xeˣ – eˣ + C. Another essential case is ∫ ln x dx, where u = ln x, dv = dx, yielding x ln x – x + C.
经典例子是 ∫ x eˣ dx：令 u = x，dv = eˣ dx，则 du = dx，v = eˣ，得到 xeˣ – ∫ eˣ dx = xeˣ – eˣ + C。另一个必会例子是 ∫ ln x dx，此时令 u = ln x，dv = dx，得出 x ln x – x + C。
Sometimes integration by parts must be applied twice, or yields the original integral again, leading to an equation that can be solved for the integral. This occurs with integrals like ∫ eˣ sin x dx, which appear in IB HL and are a good test of algebraic persistence.
有时需要两次分部积分，或者积分结果会重新出现原积分，此时可通过解方程求得积分。这常见于 ∫ eˣ sin x dx 之类的问题，常出现在 IB HL 中，是检验代数恒心的好题。
7. Partial Fractions and Rational Functions | 部分分式与有理函数积分
Integrating rational functions of the form P(x)/Q(x), where degree(P) < degree(Q), often involves expressing the fraction as a sum of simpler fractions. This technique, called partial fraction decomposition, is used extensively in IB and WJEC when integrating algebraic fractions.
对于形如 P(x)/Q(x) 的有理函数（其中 deg(P) < deg(Q)），积分时通常需要将其拆分为几个较简单的分式之和。这种称为部分分式分解的技巧在 IB 和 WJEC 有理函数积分中被广泛运用。
Common decompositions include distinct linear factors: A/(x–a) + B/(x–b); repeated linear factors: A/(x–a) + B/(x–a)²; and irreducible quadratic factors: (Ax+B)/(x²+c). Each term is then integrated using logarithms, power rules, or arctangent substitutions.
常见的分解形式有：不同线性因子，拆分为 A/(x–a) + B/(x–b)；重复线性因子，拆分为 A/(x–a) + B/(x–a)²；以及不可约二次因子，拆分为 (Ax+B)/(x²+c)。然后对每一项分别用对数、幂函数法则或反正切换元来积分。
For example, ∫ (2x+3)/((x+1)(x–2)) dx can be expressed as ∫ [1/(x+1) + 1/(x–2)] dx = ln|x+1| + ln|x–2| + C. Always check the decomposition by combining back to a common denominator.
例如 ∫ (2x+3)/((x+1)(x–2)) dx 可表示为 ∫ [1/(x+1) + 1/(x–2)] dx = ln|x+1| + ln|x–2| + C。每次分解后都要通分还原来检验是否正确。
In IB examinations, you may be required to find unknown constants by equating coefficients or substituting convenient x-values. WJEC questions are structured similarly and often lead to a combination of logs and rational integrals.
在 IB 考试中，你可能需要通过比较系数或代入特殊的 x 值来求未知常数。WJEC 的题型结构类似，通常最终会得到对数与有理积分的组合。
8. Definite Integrals and Area Under a Curve | 定积分与曲线下方面积
A definite integral computes the net signed area between a curve and the x-axis over an interval [a, b]. The Fundamental Theorem of Calculus states that ∫ₐᵇ f(x) dx = F(b) – F(a), where F is an antiderivative of f. It is vital to evaluate the top limit minus the bottom limit carefully.
定积分计算的是曲线与 x 轴在区间 [a, b] 上所围成的净符号面积。微积分基本定理指出 ∫ₐᵇ f(x) dx = F(b) – F(a)，其中 F 是 f 的一个原函数。务必仔细计算上限值减去下限值。
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) – F(a)
When the curve falls below the x-axis, the integral yields a negative value. To find the actual geometric area, split the region at x-intercepts and total the absolute values. Both IB and WJEC frequently ask for ‘the total area bounded by the curve and the x-axis’, requiring this careful partitioning.
当曲线落在 x 轴下方时，积分结果为负值。要计算实际的几何面积，需要在 x 轴截距处将区域分段，然后对绝对值求和。IB 和 WJEC 常会要求“曲线与 x 轴所围成的总面积”，这就需要进行这种谨慎的分段处理。
Area between two curves: the area between y = f(x) and y = g(x) from x = a to x = b is ∫ₐᵇ [f(x) – g(x)] dx, provided f(x) ≥ g(x) on [a, b]. Visual verification of which function is upper is helpful to avoid sign errors.
两条曲线之间的面积：从 x = a 到 x = b 之间，y = f(x) 与 y = g(x) 所夹面积为 ∫ₐᵇ [f(x) – g(x)] dx，前提是在 [a, b] 上 f(x) ≥ g(x)。通过画图判断哪条是上函数，有助于避免符号错误。
Always simplify the integrand before integrating, and be cautious with signs when substituting limits. In both syllabuses, giving the area as a negative number when a positive area is required is a frequent error.
积分前一定要先化简被积函数，代入上下限时符号要格外小心。在两种课程体系中，要求正面积时却错给出负值，是常见的失分点。
9. Volumes of Revolution | 旋转体体积
When a region is rotated around the x-axis (or y-axis), it generates a solid of revolution. The volume is calculated by integrating the cross-sectional area. For rotation about the x-axis, the formula is V = π ∫ₐᵇ [y]² dx, where y = f(x) is the boundary curve. This is a standard part of IB AI/AA and WJEC applied mathematics.
将平面区域绕 x 轴（或 y 轴）旋转一周，会生成一个旋转体。其体积通过对截面积分求得。对于绕 x 轴旋转，公式为 V = π ∫ₐᵇ [y]² dx，其中 y = f(x) 是边界曲线。这是 IB AI/AA 和 WJEC 应用数学中的标准内容。
V = π ∫ₐᵇ y² dx (rotation about x-axis)
For rotation about the y-axis, the roles of x and y are swapped: V = π ∫cd x² dy. Often you need to express x in terms of y and determine the new limits c and d from the y-coordinates. IB HL questions sometimes combine the two formulas and require the volume of a hollow solid by subtracting an inner volume.
对于绕 y 轴旋转，x 和 y 的角色互换：V = π ∫cd x² dy。通常需要将 x 用 y 表示，并根据 y 坐标确定新的积分限 c 和 d。IB HL 题目有时会结合两个公式，要求通过减去内部体积来求空心立体的体积。
Be careful to square the whole y-expression before integrating. Errors often occur when students forget the square or attempt to simplify incorrectly. Always check if the solid lies between two curves by subtracting squared radii: V = π ∫ₐᵇ [R(x)² – r(x)²] dx.
积分前务必对整个 y 表达式平方。学生常因忘记平方或化简错误而失分。当立体是由两曲线之间区域旋转而成时
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