GCSE OCR Maths: Integration | 积分考点精讲

📚 GCSE OCR Maths: Integration | 积分考点精讲

Integration is one of the two fundamental ideas in calculus, alongside differentiation. In the OCR GCSE Mathematics Higher Tier, you will learn integration as the reverse process of differentiation, and you will use it to find indefinite integrals of polynomial functions and to calculate areas under curves. Mastering integration opens the door to understanding motion, accumulation, and much more in A-level Maths. This guide covers every essential point you need for your exam, with clear explanations, worked examples, and practical tips.

积分是微积分中两个基本概念之一，与微分并列。在 OCR GCSE 数学高级级别中，你将学习积分作为微分的逆运算，并用它求多项式函数的不定积分以及计算曲线下的面积。掌握积分不仅能帮你应对考试，也为 A-level 数学中理解运动、累积量等概念打下基础。本文涵盖了考试所需的每一个关键点，配有清晰的解释、例题和实用技巧。

1. What is Integration? | 什么是积分？

Integration is a mathematical process that can be thought of as the opposite of differentiation. While differentiation finds the gradient of a curve at a point, integration finds the original function from its derivative. It can also be used to calculate the exact area bounded by a curve and the x‑axis.

积分是一种数学运算，可以看作是微分的逆运算。微分是求曲线上某一点的梯度，而积分是由导数还原出原函数。积分还可以用来计算由曲线与 x 轴所围成的精确面积。

In OCR GCSE, integration appears in two main forms: indefinite integration (finding an antiderivative) and definite integration (evaluating area over an interval). Both rely on a few straightforward rules applied to powers of x.

在 OCR GCSE 中，积分主要有两种形式：不定积分（求原函数）和定积分（求某个区间上的面积）。两者都依赖于几条针对 x 的幂的简单法则。

2. The Reverse of Differentiation | 作为微分逆运算的积分

If you are given a derivative f ‘(x), integrating it returns the original function f(x) (plus a constant). For instance, if dy/dx = 2x, then the original function y could be x², because differentiating x² gives 2x. Symbolically, ∫ 2x dx = x² + C.

如果你已知导数 f ‘(x)，对它积分就能得到原函数 f(x)（外加一个常数）。例如，如果 dy/dx = 2x，那么原函数 y 可能是 x²，因为对 x² 微分得 2x。用符号表示就是 ∫ 2x dx = x² + C。

This reverse relationship is the key to understanding integration. Every differentiation rule has a corresponding integration rule. For GCSE, we focus on the power rule.

这种互逆关系是理解积分的关键。每一条微分法则都对应着一条积分法则。在 GCSE 中，我们重点掌握幂法则。

3. The Power Rule for Integration | 幂函数的积分法则

The most important integration rule for GCSE is the power rule: to integrate xⁿ, you add 1 to the power and then divide by the new power, provided n ≠ −1.

GCSE 中最重要的积分法则是幂法则：对 xⁿ 积分时，先将指数加 1，再除以新的指数，前提是 n ≠ −1。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1

This is the exact reverse of the differentiation rule ‘multiply by the power and subtract 1 from the power’. Here, we ‘add 1 to the power and divide by the new power’.

这正好与微分的“指数乘下来，指数减 1”法则相反。积分时我们“指数加 1，除以新指数”。

Special cases: ∫ 1 dx = x + C (since 1 = x⁰), and ∫ x dx = x²/2 + C. Constant multiples can be taken outside the integral: ∫ a·f(x) dx = a ∫ f(x) dx.

特殊情况：∫ 1 dx = x + C（因为 1 = x⁰），∫ x dx = x²/2 + C。常数倍可以提到积分号外：∫ a·f(x) dx = a ∫ f(x) dx。

4. The Constant of Integration (+C) | 积分常数 (+C)

Whenever you find an indefinite integral, you must add a constant C, because differentiation removes any constant term. For example, the derivative of x² + 5 is 2x, and the derivative of x² − 3 is also 2x. So when we integrate 2x, we write x² + C to cover all possibilities.

每当你求不定积分时，都必须加一个常数 C，因为微分会消去任何常数项。例如，x² + 5 和 x² − 3 的导数都是 2x。因此，对 2x 积分时，我们写成 x² + C 来表示所有可能。

In definite integration, the +C cancels out, so it is not written. But for indefinite integration, omitting +C will lose marks in your exam.

在定积分中，+C 会互相抵消，所以不需要写出。但对不定积分，漏写 +C 会在考试中扣分。

5. Integrating Polynomials Step‑by‑Step | 多项式积分分步解析

To integrate a polynomial, integrate each term separately using the power rule and then add the results. For example, find ∫ (3x² − 4x + 5) dx.

要对多项式积分，可以逐项应用幂法则，然后把结果加起来。例如，求 ∫ (3x² − 4x + 5) dx。

Integrate 3x²: add 1 to the power → 3x³/3 = x³.

对 3x² 积分：指数加 1 → 3x³/3 = x³。
Integrate −4x: −4x²/2 = −2x².

对 −4x 积分：−4x²/2 = −2x²。
Integrate 5: 5x.

对 5 积分：5x。

Combining: ∫ (3x² − 4x + 5) dx = x³ − 2x² + 5x + C.

合并：∫ (3x² − 4x + 5) dx = x³ − 2x² + 5x + C。

Always remember to include the +C for indefinite integrals. Write your final answer in descending powers of x, as this is the standard form expected by examiners.

一定记得在不定积分末尾加上 +C。最后按 x 的降幂书写答案，这是考官期望的标准格式。

6. Definite Integration: Notation and Meaning | 定积分：符号与含义

A definite integral has limits of integration and gives a numerical value. It is written as ∫_a^b f(x) dx, where a and b are the lower and upper limits. The result represents the signed area between the curve y = f(x) and the x‑axis from x = a to x = b.

定积分带有积分上下限，计算结果是一个数值。记作 ∫_a^b f(x) dx，其中 a 和 b 分别是下限和上限。计算结果表示曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间的带符号面积。

To evaluate a definite integral, first find the indefinite integral (without +C), then substitute b and a, and subtract: F(b) − F(a).

计算定积分时，先求出不定积分（不加 +C），再代入 b 和 a 并相减：F(b) − F(a)。

7. Calculating a Definite Integral | 定积分的计算

Evaluate the definite integral ∫₁³ (2x + 1) dx.

计算定积分 ∫₁³ (2x + 1) dx。

Step 1: Integrate without +C. ∫ (2x + 1) dx = x² + x.

步骤 1：求不定积分（不加 C）。∫ (2x + 1) dx = x² + x。

Step 2: Substitute upper limit: (3)² + 3 = 9 + 3 = 12.

步骤 2：代入上限：(3)² + 3 = 9 + 3 = 12。

Step 3: Substitute lower limit: (1)² + 1 = 1 + 1 = 2.

步骤 3：代入下限：(1)² + 1 = 1 + 1 = 2。

Step 4: Subtract: 12 − 2 = 10. The exact area is 10 square units.

步骤 4：相减：12 − 2 = 10。精确面积是 10 平方单位。

This simple process works for any polynomial function. Always show clear substitution steps to gain method marks.

这个简单步骤适用于任何多项式函数。务必清晰地展示代入过程，以获得方法分。

8. Area Between a Curve and the x‑axis (Above Axis) | 曲线与 x 轴的面积（轴上方）

When the graph of y = f(x) lies above the x‑axis between x = a and x = b, the definite integral ∫_a^b f(x) dx gives the area. For example, find the area under y = x² from x = 0 to x = 2.

当 y = f(x) 的图像在 x 轴上方（从 x = a 到 x = b）时，定积分 ∫_a^b f(x) dx 直接给出面积。例如，求 y = x² 从 x=0 到 x=2 下方的面积。

Integrate: ∫ x² dx = x³/3. Evaluate from 0 to 2: (2³/3) − (0³/3) = 8/3 − 0 = 8/3. The area is 8/3 square units.

积分：∫ x² dx = x³/3。在 0 到 2 上计算：(2³/3) − (0³/3) = 8/3。面积为 8/3 平方单位。

In exam questions, you may be asked to shade the area and then calculate it. Always draw a quick sketch to check whether the curve is above the axis.

考试中可能要求你先给区域涂上阴影，再计算面积。画一个简图快速判断曲线是否在轴上方，是一个好习惯。

9. Handling Area When the Curve Goes Below the x‑axis | 曲线位于 x 轴下方时的面积处理

If part of the curve lies below the x‑axis, the definite integral for that part will be negative. Area must always be positive, so you need to split the integral at the roots and take the absolute value of each negative portion.

如果曲线的一部分位于 x 轴下方，该部分的定积分为负值。面积必须是正的，因此需要在根处将积分分开，并对负值部分取绝对值。

Example: find the total area enclosed by y = x(x − 2) and the x‑axis from x = 0 to x = 2. Note the curve is below the axis for 0 < x < 2. Integrating directly gives a negative number; instead compute ∫₀² x(x−2) dx = −4/3, so the area is │−4/3│ = 4/3.

例题：求 y = x(x − 2) 与 x 轴在 0 到 2 围成的总面积。注意在 0<x<2 区间曲线在轴下方。直接积分会得负数；应计算 ∫₀² x(x−2) dx = −4/3，取绝对值，面积为 4/3。

Alternatively, split the integral into regions where the function is positive and negative, evaluate each separately, and add their absolute values. This is a common GCSE trap – always check a sketch.

或者，将积分按函数正负分段，分别计算并取绝对值再相加。这是 GCSE 常见的陷阱——一定要画图检查。

10. Exam Tips for OCR GCSE Integration Questions | OCR GCSE 积分考题技巧

Learn to recognise when to integrate and when to differentiate. Integration is often signalled by words like ‘find the original equation’, ‘area under the curve’, or the integral symbol.

学会辨别什么时候积分、什么时候微分。题目中出现“求原方程”“曲线下方面积”或积分符号时，通常要用积分。
Always include dx (or the appropriate differential) in indefinite integrals and at the end of definite integrals. Missing it can cost marks.

永远不要漏写 dx（或相应的微分符），不定积分和定积分末端都要写。漏写可能丢分。
Double-check your integration by differentiating your answer. The result should be the original integrand. This is a powerful self-check.

用微分检验积分结果：对你的答案求导，应该得到原来的被积函数。这是非常有用的自检方法。
When a question asks for the area under a curve, draw a quick sketch to see if the curve crosses the axis. If it does, split the area.

题目要求曲线下方面积时，画个简图判断曲线是否穿过 x 轴。如果穿过，就要分段计算面积。
In longer questions, show all your substitution steps clearly. Even if your final answer is wrong, you can earn method marks for correct integration and substitution.

在长题目中，清楚地展示每一步代入过程。即使最终答案错误，正确的积分和代入过程也能获得方法分。

With regular practice of past OCR papers, integration will become a reliable source of marks. Treat it as a set of simple steps – increase the power, divide by the new power, apply limits, and handle signs.

通过定期练习 OCR 历年真题，积分会成为你稳定得分的题目。把它当作一套简单步骤：指数加 1，除以新指数，代入上下限，处理正负。

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GCSE OCR Maths: Integration | 积分考点精讲