📚 A-Level OCR Maths: Integration Key Points | A-Level OCR 数学：积分 考点精讲
Integration is the reverse process of differentiation and is a cornerstone of A-Level OCR Mathematics. It allows us to find areas under curves, solve differential equations, and model physical quantities such as displacement and velocity. This article distils the essential integration techniques and applications you need for your exams, with clear explanations, worked examples, and practical tips.
积分是微分的逆运算，也是 A-Level OCR 数学的核心内容。它帮助我们求解曲线下方面积、解微分方程，以及建立位移、速度等物理量的模型。本文提炼了考试必备的积分技巧与应用，配以清晰的讲解、实例和实用建议，助你高效备考。
1. Introduction to Integration | 积分简介
Integration is the process of finding a function from its derivative. If F'(x) = f(x), then F(x) is an antiderivative of f(x), and we write ∫ f(x) dx = F(x) + C, where C is an arbitrary constant. This is called indefinite integration because no limits are specified.
积分是由导数寻求原函数的过程。若 F'(x) = f(x)，则 F(x) 是 f(x) 的一个原函数，记作 ∫ f(x) dx = F(x) + C，其中 C 为任意常数。由于没有指定上下限，这称为不定积分。
Integration reverses differentiation, so every differentiation rule gives rise to an integration rule. For example, since the derivative of xⁿ is nxⁿ⁻¹, the integral of xⁿ is xⁿ⁺¹/(n+1) (for n ≠ -1).
积分是微分的逆运算，因此每条微分法则都可以推导出一条积分法则。例如，由于 xⁿ 的导数是 nxⁿ⁻¹，所以 xⁿ 的积分是 xⁿ⁺¹/(n+1)（n ≠ -1）。
In OCR papers, you are expected to recognise the need for integration from the context, such as finding the area under a curve or recovering a function from a given gradient.
在 OCR 考试中，你需要根据上下文识别需要积分的场景，例如求曲线下方面积或从已知斜率恢复原函数。
2. Indefinite Integration and Basic Rules | 不定积分与基本规则
The general power rule for integration is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1. This rule is fundamental and must be applied correctly, especially for fractional and negative powers.
积分的一般幂法则为 ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，其中 n ≠ -1。这条法则是基础，必须正确应用，尤其对于分数幂与负幂。
Standard integrals you must memorise include:
必须熟记的标准积分包括：
- ∫ eˣ dx = eˣ + C
- ∫ 1/x dx = ln|x| + C
- ∫ sin x dx = -cos x + C
- ∫ cos x dx = sin x + C
- ∫ sec² x dx = tan x + C
- ∫ cosec x cot x dx = -cosec x + C
- ∫ sec x tan x dx = sec x + C
- ∫ cosec² x dx = -cot x + C
For linear functions: ∫ f(ax + b) dx = (1/a) F(ax + b) + C, where F is an antiderivative of f. This reverse chain rule is crucial for efficiency.
对于线性函数：∫ f(ax + b) dx = (1/a) F(ax + b) + C，其中 F 是 f 的一个原函数。这种逆链式法则对提高解题效率至关重要。
3. Definite Integration and the Area Under a Curve | 定积分与曲线下方面积
A definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve y = f(x), the x-axis, and the lines x = a and x = b. The value is evaluated using the fundamental theorem of calculus: ∫ₐᵇ f(x) dx = F(b) – F(a), where F'(x) = f(x).
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 之间所夹的有符号面积。其值由微积分基本定理计算：∫ₐᵇ f(x) dx = F(b) – F(a)，其中 F'(x) = f(x)。
Crucially, areas below the x-axis yield negative values. To find the true geometric area, you must split the integral at the points where the curve crosses the x-axis and take the absolute value of each part.
关键点在于，x 轴下方的区域积分结果为负值。要计算真正的几何面积，必须在曲线穿越 x 轴的点处拆分积分，并对每一段取绝对值。
Area = ∫ₐᵇ |f(x)| dx
Always sketch the curve first to identify regions and avoid sign errors.
务必先画出曲线的示意图，以识别区域并避免符号错误。
4. Area Between Two Curves | 两曲线间的面积
When the region is bounded by two curves y = f(x) and y = g(x) from x = a to x = b, the enclosed area is ∫ₐᵇ [upper curve – lower curve] dx.
当区域由两条曲线 y = f(x) 和 y = g(x) 在 x = a 到 x = b 之间围成时，所围的面积为 ∫ₐᵇ [上方曲线 – 下方曲线] dx。
Identify which function is on top over the interval; if they cross, split the integral at the intersection points.
首先要判断在该区间内哪条曲线在上方；如果它们相交，则需要在交点处拆分积分。
For example, to find the area enclosed between y = x² and y = 2 – x², find intersection points by solving x² = 2 – x², giving x = ±1. Then integrate (2 – x²) – x² from -1 to 1.
例如，求 y = x² 与 y = 2 – x² 所围成的面积，先解方程 x² = 2 – x² 得交点 x = ±1，然后从 -1 到 1 对 (2 – x²) – x² 积分。
Area = ∫₋₁¹ (2 – 2x²) dx
5. Integration by Substitution | 代换积分法
Substitution is the reverse of the chain rule. Choose u = g(x) such that du/dx appears in the integrand, then replace dx with du/(du/dx).
代换法是链式法则的逆运算。选取 u = g(x)，使得被积函数中出现 du/dx，然后用 du/(du/dx) 替换 dx。
For definite integrals, you must change the limits from x-values to u-values when substituting. This avoids switching back to x and is less error-prone.
对于定积分，代换时必须将上下限从 x 值转换为 u 值，这样无需再换回 x，也能减少错误。
Example: ∫ 2x√(x²+1) dx. Let u = x²+1, then du/dx = 2x, so dx = du/(2x). The integral becomes ∫ √u du = (2/3) u³/² + C. Replace u with x²+1 to finish.
例如：∫ 2x√(x²+1) dx。令 u = x²+1，则 du/dx = 2x，dx = du/(2x)。积分变为 ∫ √u du = (2/3) u³/² + C，最后将 u 换回 x²+1。
Common substitutions include u = ax + b for linear inner functions, u = sin x or u = cos x for trigonometric products, and u = f(x) for exponential or logarithmic integrals.
常见的代换包括：线性内层函数用 u = ax + b；三角函数乘积用 u = sin x 或 u = cos x；指数或对数积分用 u = f(x)。
6. Integration by Parts | 分部积分法
Integration by parts comes from the product rule for differentiation. The formula is ∫ u dv = uv – ∫ v du, or in the form given in the formula booklet: ∫ u (dv/dx) dx = uv – ∫ v (du/dx) dx.
分部积分法源自微分的乘法法则，公式为 ∫ u dv = uv – ∫ v du，或在公式册中给出的形式：∫ u (dv/dx) dx = uv – ∫ v (du/dx) dx。
Choose u according to LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) – earlier functions in the list are normally chosen as u. Common choices are u = ln x, u = x, or u = polynomial.
根据 LIATE 法则选择 u（对数函数、反三角函数、代数函数、三角函数、指数函数），列表中优先出现的函数通常选作 u。常见的 u 有 ln x、x 或多项式。
Example: ∫ x eˣ dx. Let u = x, dv/dx = eˣ, so du/dx = 1, v = eˣ. Then ∫ x eˣ dx = x eˣ – ∫ eˣ dx = x eˣ – eˣ + C.
例如：∫ x eˣ dx。令 u = x, dv/dx = eˣ，则 du/dx = 1, v = eˣ。于是 ∫ x eˣ dx = x eˣ – ∫ eˣ dx = x eˣ – eˣ + C。
Sometimes you need to apply integration by parts twice, or recognise that the original integral reappears, allowing algebraic rearrangement (e.g., ∫ eˣ sin x dx).
有时需要两次使用分部积分法，或者注意到原积分再次出现，通过代数整理即可求解（如 ∫ eˣ sin x dx）。
7. Integrating Parametric Equations | 参数方程积分
When a curve is defined parametrically by x = f(t), y = g(t), the area under the curve between t = t₁ and t = t₂ is given by ∫ y (dx/dt) dt. The limits of integration are the parameter values, not x-values.
当曲线由参数方程 x = f(t), y = g(t) 给出时，曲线下方在 t = t₁ 到 t = t₂ 之间的面积由 ∫ y (dx/dt) dt 给出。积分的上下限是参数 t 的值，而非 x 值。
The alternative form ∫ x (dy/dt) dt can be used depending on what is convenient. Both expressions come from the chain rule: ∫ y dx = ∫ y (dx/dt) dt.
另一种形式 ∫ x (dy/dt) dt 也可根据便利性使用。这两种表达式均由链式法则得到：∫ y dx = ∫ y (dx/dt) dt。
Always check the orientation of the parameter: if t increases, the integration goes from a smaller t to a larger t. If the curve traces backwards, the integral may be negative, and you must take absolute values for area.
务必检查参数的走向：若 t 增加，积分方向为从小 t 到大 t。如果曲线沿相反方向描绘，积分可能为负，计算面积时需取绝对值。
8. Volumes of Revolution | 旋转体体积
The volume generated when the region under y = f(x) from x = a to x = b is rotated completely about the x-axis is V = π ∫ₐᵇ [f(x)]² dx. This formula is provided in the OCR booklet.
将 y = f(x) 下方、x = a 到 x = b 之间的区域绕 x 轴旋转一周所生成的体积为 V = π ∫ₐᵇ [f(x)]² dx。该公式在 OCR 公式册中提供。
For rotation about the y-axis, rearrange the function to x = g(y) and use V = π ∫ₛᵈ [g(y)]² dy with limits on y. Alternatively, for parametric curves, V = π ∫ₜ₁ᵗ² y² (dx/dt) dt when rotating about the x-axis.
若绕 y 轴旋转，需将函数改写为 x = g(y)，并使用 V = π ∫ₛᵈ [g(y)]² dy，其上下限为 y 值。对于参数曲线，绕 x 轴旋转时则有 V = π ∫ₜ₁ᵗ² y² (dx/dt) dt。
Beware of composite regions: subtract volumes if there is a hollow part (washer method). Sketch the region and its rotation to visualise the solid.
注意复合区域：若存在空心部分，需减去相应体积（垫圈法）。画出区域及其旋转后的示意图，有助于想象立体图形。
9. Differential Equations: Separation of Variables | 微分方程：分离变量法
OCR A-Level requires solving first-order separable differential equations of the form dy/dx = g(x)h(y). Rearrange to ∫ (1/h(y)) dy = ∫ g(x) dx, integrate both sides, and include one constant of integration.
OCR A-Level 要求求解一阶可分离变量的微分方程，形式为 dy/dx = g(x)h(y)。将其整理为 ∫ (1/h(y)) dy = ∫ g(x) dx，两边积分，并加入一个积分常数。
After integration, use given initial conditions to find the particular solution. The constant is usually added to the x-side for simplicity.
积分后，利用已知的初始条件求出特解。为简便起见，常数通常加在 x 一侧。
Example: dy/dx = 2xy, with y(0) = 3. Separate: ∫ (1/y) dy = ∫ 2x dx ⇒ ln|y| = x² + C ⇒ y = A eˣ². Using y(0)=3 gives A=3, so y = 3eˣ².
例如：dy/dx = 2xy，且 y(0) = 3。分离变量：∫ (1/y) dy = ∫ 2x dx ⇒ ln|y| = x² + C ⇒ y = A eˣ²。利用 y(0)=3 得 A=3，故 y = 3eˣ²。
Exponential growth and decay, cooling, and population models are common contexts for differential equations.
指数增长与衰减、冷却模型以及种群模型是微分方程的常见应用场景。
10. Applications in Kinematics | 运动学中的应用
In mechanics, velocity v is the integral of acceleration a with respect to time, and displacement s is the integral of velocity: v = ∫ a dt, s = ∫ v dt. Each integration introduces a constant determined by initial conditions.
在力学中，速度 v 是加速度 a 对时间 t 的积分，位移 s 是速度对时间的积分：v = ∫ a dt，s = ∫ v dt。每次积分都会引入一个由初始条件确定的常数。
For rectilinear motion, the definite integral of velocity from t₁ to t₂ gives the change in displacement; the definite integral of speed (absolute value of velocity) gives the total distance travelled.
对于直线运动，速度从 t₁ 到 t₂ 的定积分给出位移的变化量；而速率（速度的绝对值）的定积分则给出总路程。
Graphical interpretation: the area under a velocity–time graph equals displacement; the area under an acceleration–time graph equals the change in velocity.
图形解释：速度-时间图下方面积等于位移；加速度-时间图下方面积等于速度变化量。
OCR questions often combine integration with vectors, requiring you to integrate i and j components separately.
OCR 考题常将积分与向量结合，要求分别对 i 和 j 分量进行积分。
11. Trapezium Rule | 梯形法则
When an integral cannot be evaluated analytically, or you must estimate an area from data, the trapezium rule provides a numerical approximation. For n strips of width h = (b – a)/n, the approximate integral is ∫ₐᵇ y dx ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ].
当积分无法用解析方法求值，或需要根据数据估算面积时，梯形法则给出数值近似。对于 n 个宽度为 h = (b – a)/n 的条带，近似积分为 ∫ₐᵇ y dx ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]。
This formula is in the OCR booklet, but you must be able to apply it confidently. The more strips you use, the better the approximation. You may also be asked to comment on whether the trapezium rule gives an over- or under-estimate based on the curve’s concavity.
此公式在 OCR 公式册中给出，但你必须能熟练应用。条带数量越多，近似程度越好。你还可能被要求根据曲线凹凸性判断梯形法则的估算是偏大还是偏小。
For curves that are convex (curving upwards), the trapezium rule overestimates; for concave curves, it underestimates. Draw a sketch to check.
对于凸曲线（向上弯曲），梯形法则会高估值；对于凹曲线，则会低估。画出示意图即可判断。
12. Common Mistakes and Exam Tips | 常见错误与应试技巧
Always add the constant +C for indefinite integrals; omitting it often costs a mark. For definite integrals, write the result with brackets: [F(x)] with limits, then substitute.
不定积分务必加上常数 +C；遗漏通常会被扣分。对于定积分，应写出带括号的形式 [F(x)] 并标注上下限，再代入数值。
Don’t forget the modulus sign in ∫ 1/x dx = ln|x| + C unless the domain is known to be positive. Similarly, when integrating rational functions after partial fractions, include the absolute values correctly.
除非定义域已知为正，否则 ∫ 1/x dx 必须写成 ln|x| + C。同样，通过分部分式积分有理函数时，也要正确包含绝对值。
When an area crosses the x-axis, separate and use absolute values. Many students lose marks by blindly applying limits without checking for sign changes.
当区域跨越 x 轴时，需分割并取绝对值。许多学生因不问符号变化就盲目代入上下限而失分。
Practice recognising the structure of integrals: can you use substitution, parts, or is it a standard form? Look for a function and its derivative, or a product where one factor resembles the derivative of the other.
练习识别积分结构：能用代换法、分部积分法，还是标准形式？注意寻找函数与其导数的组合，或乘积中一个因子类似另一因子导数的情形。
Use your formula booklet strategically; knowing which formulas are provided saves memorisation effort, but you must know how to apply them accurately under timed conditions.
有策略地使用公式册：清楚哪些公式已提供可节省记忆负担，但你仍需在限时条件下准确运用它们。
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