IGCSE OCR Maths: Integration Key Points | IGCSE OCR 数学：积分考点精讲

📚 IGCSE OCR Maths: Integration Key Points | IGCSE OCR 数学：积分考点精讲

Integration is the fundamental process of finding the total change from a rate of change. In the IGCSE OCR syllabus, mastering integration means understanding it both as the reverse of differentiation and as a tool for calculating areas under curves. This article covers all the key concepts, notation, techniques, and exam tips you need to succeed.

积分是根据变化率求总变化的基本过程。在 IGCSE OCR 考试大纲中，掌握积分既要理解它是微分的逆运算，也要理解它是计算曲线下面积的有力工具。本文涵盖了你需要掌握的所有关键概念、符号、技巧和考试提示。

1. Introduction to Integration | 积分简介

Integration is one of the two central branches of calculus, alongside differentiation. While differentiation finds the instantaneous rate of change (gradient) of a function, integration accumulates these small changes to recover the original function or to compute total quantities like area, distance, or population growth.

积分是微积分的两大核心分支之一，另一个是微分。微分求出函数的瞬时变化率（斜率），而积分则累积这些微小变化以恢复原函数，或计算总面积、距离或种群增长等总量。

In IGCSE OCR mathematics, you will primarily meet indefinite integration (finding antiderivatives) and definite integration (calculating areas and numerical approximations). The concept of integration as an infinite sum of infinitesimal strips is key to visualising the area problem.

在 IGCSE OCR 数学中，你主要会接触到不定积分（求原函数）和定积分（计算面积和数值近似）。将积分理解成无穷多个无穷小长条面积的和，是可视化面积问题的关键。

The integral sign ∫ represents an elongated S, standing for ‘sum’. When we integrate, we are effectively summing the values of a function multiplied by tiny changes in the variable.

积分符号 ∫ 代表一个拉长的 S，表示 “和”。积分时，实际上是将函数值乘以变量的微小变化后累加起来。

2. The Reverse of Differentiation | 微分的逆运算

The most basic way to understand integration is as the opposite process to differentiation. If the derivative of F(x) is f(x), then the integral of f(x) is F(x) plus a constant. This is why integration is often called antidifferentiation.

理解积分最基础的方式，就是把它看作微分的逆过程。如果 F(x) 的导数是 f(x)，那么 f(x) 的积分就是 F(x) 加上一个常数。因此积分常被称为逆微分。

For example, if d/dx (x²) = 2x, then integrating 2x gives x². However, any constant added to x² also differentiates to 2x, so the indefinite integral includes an arbitrary constant, usually denoted by C.

例如，d/dx (x²) = 2x，那么对 2x 积分得到 x²。但是，任何常数加在 x² 上，微分后仍是 2x，因此不定积分包含一个任意常数，通常用 C 表示。

Checking your result by differentiation is a fast and reliable verification method in the exam. Always mentally differentiate your answer to ensure you recover the original integrand.

通过微分来检验你的结果是考试中快速且可靠的验证方法。永远在心里对你的答案求导，以确保得到原来的被积函数。

3. Indefinite Integral Notation | 不定积分符号

The indefinite integral of a function f(x) with respect to x is written as ∫ f(x) dx. The symbol dx indicates the variable of integration and also reminds us of the small width of each strip in an area sum.

函数 f(x) 对 x 的不定积分写作 ∫ f(x) dx。符号 dx 表示积分变量，同时也让我们联想到面积和中每个小条的宽度。

The integral sign and the differential dx act as a pair of brackets; the expression between them is the integrand. You must always include dx in your working to correctly specify the variable.

积分号与微分 dx 如同括号一样成对出现；它们之间的表达式是被积函数。你在解题时一定要写上 dx，以正确指明积分变量。

For an indefinite integral, the answer always includes ‘+ C’, where C is an arbitrary real constant. Missing the constant of integration is one of the most common mistakes in IGCSE exams.

对于不定积分，答案总是包含 “+ C”，其中 C 是一个任意实常数。漏掉积分常数是 IGCSE 考试中最常见的错误之一。

4. Integrating Power Functions | 幂函数积分

The power rule for integration is the reverse of the power rule for differentiation. For any real number n ≠ −1, the integral of xⁿ is xⁿ⁺¹/(n+1) + C.

幂函数积分法则是幂函数微分法则的逆运算。对于任何实数 n ≠ −1，xⁿ 的积分是 xⁿ⁺¹/(n+1) + C。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1

This formula is derived by raising the power by 1 and then dividing by the new power. Memorising this sequence — add one to the power, divide by the new power — will help you avoid errors.

这个公式是将指数加 1，再除以新的指数得到的。记住这个顺序——“指数加 1，除以新指数”——将帮助你避免错误。

Special care must be taken when integrating expressions like 1/x or x⁻¹, because the rule does not apply. For IGCSE OCR, you only need to know that ∫ x⁻¹ dx = ln|x| + C, although this appears rarely in core papers and may be supplied if needed.

在积分像 1/x 或 x⁻¹ 这样的表达式时要特别小心，因为该法则不适用。在 IGCSE OCR 中，你只需要知道 ∫ x⁻¹ dx = ln|x| + C，尽管这在核心试卷中很少出现，需要时可能会给出。

When integrating constant terms, remember that a constant k can be written as kx⁰, so ∫ k dx = kx + C. This simple case is often overlooked.

积分常数项时，记住常数 k 可以看作 kx⁰，所以 ∫ k dx = kx + C。这个简单情况常被忽略。

5. Integrating Polynomials Term by Term | 多项式逐项积分

Integration is a linear operation, meaning you can integrate a sum term by term and pull constant factors outside the integral sign. This makes integrating polynomials straightforward once the power rule is mastered.

积分是线性运算，意味着你可以逐项积分，并可将常数因子提到积分号外。这使得掌握了幂法则后，积分多项式变得十分直接。

∫ [a·f(x) + b·g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx

To integrate a polynomial like 3x² + 2x − 5, integrate each term individually: 3·(x³/3) + 2·(x²/2) − 5x, then simplify to x³ + x² − 5x + C. Always simplify coefficients to reduce the risk of arithmetic errors.

要积分像 3x² + 2x − 5 这样的多项式，只需逐项积分：3·(x³/3) + 2·(x²/2) − 5x，然后化简为 x³ + x² − 5x + C。务必化简系数以降低算术错误的风险。

When expressions have negative or fractional powers, first rewrite them using index laws. For example, 1/x² becomes x⁻², and √x becomes x½. Then apply the standard power rule.

当表达式含有负指数或分数指数时，首先用指数律重写。例如 1/x² 写为 x⁻²，√x 写为 x½。然后应用标准幂法则。

Algebraic simplification before integrating saves time and avoids mistakes. In an exam, you should always present the integrand in its simplest index form before proceeding.

积分前进行代数化简既省时又避免错误。在考试中，应先将被积函数化为最简单的指数形式再继续。

6. The Constant of Integration | 积分常数

The constant of integration, +C, represents an infinite family of functions that differ only by a vertical shift. Since differentiation obliterates constants, any arbitrary constant is allowed in the antiderivative.

积分常数 +C 表示一个只相差垂直平移的无限函数族。由于微分会消去常数，原函数中可以出现任意常数。

If an exam question gives an additional condition, such as a specific point through which the curve passes, you can determine the unique value of C. Substitute the coordinates and solve to find the constant.

如果考题给出了附加条件，例如曲线经过某个特定点，你就可以求出 C 的唯一值。代入坐标并解方程即可找到常数。

For example, if dy/dx = 2x and the curve passes through (1, 4), integrating gives y = x² + C. Substituting x=1, y=4 yields C=3, so the particular solution is y = x² + 3.

例如，若 dy/dx = 2x，且曲线经过 (1, 4)，积分得到 y = x² + C。代入 x=1, y=4 得 C=3，因此特解为 y = x² + 3。

Always write ‘+ C’ as soon as you perform an indefinite integration, even if you plan to find it later. This habit prevents loss of marks for missing constants.

每次进行不定积分时，即使你打算稍后再求出常数，也要立即写上 “+ C”。这个习惯可防止因漏掉常数而失分。

7. Definite Integration and Limits | 定积分与积分限

A definite integral has upper and lower limits and represents the net accumulation of a quantity between those two bounds. The notation ∫ₐᵇ f(x) dx reads as ‘the integral from a to b of f(x) dx’.

定积分有上下限，表示在两个界限之间某个量的净累积。符号 ∫ₐᵇ f(x) dx 读作 “f(x) 从 a 到 b 的积分”。

To evaluate a definite integral, first find an antiderivative F(x) without the constant, then compute F(b) − F(a). This is the Fundamental Theorem of Calculus, connecting differentiation and integration.

要计算定积分，先求出不含常数的原函数 F(x)，然后计算 F(b) − F(a)。这就是微积分基本定理，它将微分与积分联系起来。

∫ₐᵇ f(x) dx = F(b) − F(a)

Use square brackets to show the substitution step: [F(x)]ₐᵇ. This structured working helps you track signs and avoid careless errors when substituting the lower limit.

用方括号表示代入步骤：[F(x)]ₐᵇ。这种有条理的书写可帮助你跟踪符号，避免在代入下限时粗心出错。

Definite integrals can yield negative values, which in area problems indicates that the region lies below the x-axis. Do not assume a negative result is automatically wrong.

定积分可能得到负值，这在面积问题中表示区域位于 x 轴下方。不要假定负值结果就一定是错的。

8. Area Under a Curve | 曲线下的面积

One of the most important applications of integration is finding the area bounded by a curve y = f(x), the x-axis, and the vertical lines x = a and x = b. The area is given by the definite integral ∫ₐᵇ f(x) dx, provided f(x) ≥ 0 over the interval.

积分最重要的应用之一，就是求曲线 y = f(x)、x 轴以及垂直线 x = a 和 x = b 围成的面积。如果在该区间内 f(x) ≥ 0，面积由定积分 ∫ₐᵇ f(x) dx 给出。

If the graph dips below the x-axis, you must split the area into separate parts where f(x) is positive and where it is negative. Compute the positive area segments separately and take absolute values before summing.

如果图像延伸到 x 轴下方，你必须将面积分成 f(x) 为正和为负的独立部分。分别计算正面积部分，取绝对值后再求和。

Always sketch the graph or identify the x‑intercepts within the interval to determine where the function changes sign. Exam questions often ask for the total area enclosed, not net signed area.

一定要画草图或确定区间内的 x 截距，以判断函数在哪里变号。考题常要求求封闭的总面积，而不是带符号的净值面积。

The units of area are square units and must be stated if the question requires a context‑based answer. Even in non‑context problems, showing the working clearly is essential for earning method marks.

面积单位是平方单位，如果题目要求基于情境的回答，必须注明。即使在非情境问题中，清晰地展示解题过程对获取方法分也至关重要。

9. Area Between a Curve and the x-axis | 曲线与x轴之间的面积

To find the total area between a curve and the x-axis from a to b, first solve f(x)=0 to find any roots inside [a,b]. These roots are the points where the curve crosses the axis and indicate where the area must be split.

要求出从 a 到 b 之间曲线与 x 轴的总面积，首先解方程 f(x)=0 求出 [a,b] 内的所有根。这些根是曲线与轴的交点，指示出必须在何处分割面积。

Do not simply integrate from a to b and take the absolute value of the result. That yields signed net area, which can be much smaller than the true total area if the curve crosses the axis.

不要简单地计算从 a 到 b 的积分再取绝对值，那样会得到带有符号的净面积，如果曲线跨过轴，这个值可能远小于真实的全面积。

During exam revision, set up a table for the signs of f(x) in each subinterval to systematise your approach. This method reduces the mental load and prevents sign mistakes.

在复习备考时，为每个子区间内 f(x) 的符号制一张表格，可使解题步骤系统化。这种方法能减轻思维负担，防止符号错误。

10. Area Between Two Curves | 两曲线之间的面积

The area enclosed between two curves y = f(x) and y = g(x) from x=a to x=b is calculated by integrating the difference of the functions: Area = ∫ₐᵇ [f(x) − g(x)] dx, provided f(x) ≥ g(x) throughout the interval.

两条曲线 y = f(x) 与 y = g(x) 在 x=a 到 x=b 之间围成的面积，通过积分两函数之差来计算：面积 = ∫ₐᵇ [f(x) − g(x)] dx，前提是在整个区间内 f(x) ≥ g(x)。

If the curves intersect, find the intersection points and split the interval accordingly. In each subinterval, identify which curve is on top and integrate the difference (top minus bottom).

如果两曲线相交，则求出交点并相应分割区间。在每个子区间内，确定哪条曲线在上方，然后积分差值（上方减下方）。

Draw a clear sketch and shade the region of interest. Labelling which function represents the upper boundary and which the lower boundary will clarify the setup for your integral.

画一个清晰的草图，并给感兴趣的区域涂上阴影。标出哪个函数表示上边界、哪个表示下边界，可以使你的积分设定更加清晰。

A typical IGCSE OCR exam question might ask for the area between a line and a parabola. Setting up the integral correctly and finding the intersection limits are the major assessed skills.

一道典型的 IGCSE OCR 考试题目可能会要求求一条直线与一条抛物线之间的面积。正确设定积分式并求出交点界限是主要考查的技能。

11. The Trapezium Rule | 梯形法则

When exact integration is impossible or when only a table of data is given, the Trapezium Rule provides an approximate value for a definite integral. It works by splitting the area into a number of trapezoids of equal width h.

当无法精确积分或只给出数据表格时，梯形法则可以为定积分提供一个近似值。它的原理是将面积分割成若干个等宽 h 的梯形。

∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]

Here, y₀ = f(a), yₙ = f(b), and the yᵢ are the function values at equally spaced points. The width h = (b − a)/n, where n is the number of strips (trapezia).

这里，y₀ = f(a)，yₙ = f(b)，而 yᵢ 是等距点上的函数值。宽度 h = (b − a)/n，其中 n 是条带（梯形）的数量。

Increasing the number of strips generally improves the accuracy of the approximation. You should be prepared to comment on whether the trapezium rule gives an overestimate or underestimate by considering the shape of the curve.

增加条带数通常可以提高近似的精度。你应准备好通过考虑曲线形状，来评论梯形法则是给出高估值还是低估值。

For a concave curve that lies above its chords, the trapezium rule overestimates the true area; for a convex curve, it underestimates. This insight is often tested in exam reasoning questions.

对于位于其弦上方的凹曲线，梯形法则会高估真实面积；对于凸曲线，则会低估。这种洞察常在考试的说理题中考查。

12. Summary and Exam Tips | 总结与考试技巧

Success in IGCSE OCR integration questions relies on fluency with the power rule, careful handling of constants and limits, and the ability to interpret area problems correctly. Regular practice writing all steps neatly will build confidence.

在 IGCSE OCR 积分题中取得成功，依赖于对幂法则的熟练掌握、对常数和积分限的仔细处理，以及正确解读面积问题的能力。经常整齐地书写所有步骤的练习将建立信心。

Remember the four golden rules: raise the power by one and divide by the new power; always add +C for indefinite integrals; use F(b)−F(a) for definite integrals; and split areas at any x‑intercept.

记住四条金律：指数加一后除以新指数；不定积分永远加 +C；定积分使用 F(b)−F(a)；在任何 x 截距处分割面积。

When revising, create a quick‑reference card with the integration power rule, special forms like ∫ 1/x dx, and the trapezium rule formula. Familiarity with these will save time under exam pressure.

复习时，制作一张快速参考卡，写上积分幂法则、∫ 1/x dx 这样的特殊形式以及梯形法则公式。对它们的熟悉能在考试压力下节省时间。

Always check your answer by differentiating the antiderivative. This simple habit can instantly reveal algebraic slips, especially with signs and coefficients, giving you a chance to correct them before moving on.

永远通过对原函数求导来检验你的答案。这个简单习惯能立即揭示代数失误，尤其是符号和系数方面的错误，让你有机会在继续做下一题前改正。

Finally, engage with past papers and mark schemes to see how marks are allocated. Often, even an incorrect numerical answer can secure most of the method marks if your working is clear and logical.

最后，要研读历年真题及评分标准，了解分值是如何分配的。很多时候，即使数值答案错了，只要解题步骤清晰、逻辑通顺，也能拿到绝大部分的方法分。

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IGCSE OCR Maths: Integration Key Points | IGCSE OCR 数学：积分 考点精讲