Integration Dominoes: A-Level Maths Integration Worksheet Activities Explained | A-Level 数学：积分多米诺骨牌练习题型解析

📚 Integration Dominoes: A-Level Maths Integration Worksheet Activities Explained | A-Level 数学：积分多米诺骨牌练习题型解析

Integration dominoes have become a popular classroom and self-study resource in A-Level Mathematics. They transform routine integration practice into an engaging puzzle where each ‘domino’ links an integral to its antiderivative. This article explores how these worksheets work, the integration techniques they reinforce, and how you can use them to sharpen your skills for the exam.

积分多米诺骨牌已成为A-Level数学中广受欢迎的课堂与自学资源。它们把常规的积分练习转化为富有吸引力的拼图，每一张”骨牌”都将一个积分与它的原函数连接起来。本文探讨这些练习纸的运作方式、它们所强化的积分技巧，以及你如何利用它们为考试磨练技能。

1. What Is an Integration Domino? | 什么是积分多米诺？

An integration domino is a puzzle activity based on the classic domino game. Each tile contains an integral on its left side and an antiderivative (or an expression) on its right side. The goal is to arrange the tiles end‑to‑end so that the antiderivative on one tile matches the integral on the next tile, forming a continuous loop or a linear chain.

积分多米诺是一种基于经典多米诺骨牌游戏的拼图活动。每张骨牌左侧写有一个积分，右侧写有一个原函数（或表达式）。目标是将骨牌首尾相连地排列，使一张骨牌上的原函数与下一张骨牌上的积分配对，形成一个闭合回路或一条线性链。

For example, a tile could show ∫ 3x² dx on the left and x³ + C on the right. The next tile must then start with an integral whose integrand is related to x³ + C, or perhaps a completely different integral that equals x³ + C after adjustment. This forces you to think backwards and forwards, reinforcing the fundamental theorem of calculus.

例如，一张骨牌左侧可以显示 ∫ 3x² dx，右侧显示 x³ + C。那么下一张骨牌必须以一个与 x³ + C 关联的被积函数开头，或者是一个经调整后结果等于 x³ + C 的完全不同的积分。这迫使你既正向又逆向思考，从而强化微积分基本定理。

2. How Domino Worksheets Work | 多米诺练习纸如何运作

A typical domino worksheet provides a set of jumbled tiles, often printed on a single sheet, which students cut out or reorder on paper. Starting from a tile labelled ‘Start’, you work out the integral on its right side (if that is the given answer format) or match answers. In some versions, each tile’s left side is a question, and the right side is an answer to a different question. The puzzle is complete when all tiles form a closed shape or a predetermined path.

一张典型的多米诺练习纸会提供一组打乱顺序的骨牌，通常印刷在同一张纸上，由学生裁剪或在纸上重新排序。从一张标有”Start”的骨牌开始，你需要计算出其右侧的积分结果（如果采用这种答案格式）或进行匹配。在有些版本中，每张骨牌的左侧是一个问题，右侧是另一个问题的答案。当所有骨牌组成一个封闭形状或预设路径时，拼图即告完成。

These worksheets can be tailored to any integration topic: basic power rule, trigonometric integrals, exponential functions, substitution, integration by parts, and even partial fractions. They are particularly effective because an error in one tile prevents the domino chain from linking correctly, giving immediate feedback without needing a mark scheme.

这些练习纸可以针对任何积分主题定制：基本幂法则、三角函数积分、指数函数、代换法、分部积分法，甚至部分分式。它们特别有效，因为一张骨牌上的错误会让多米诺链无法正确连接，从而无需评分方案即可立即得到反馈。

3. Key Integration Techniques Needed | 必要积分技巧

To succeed with integration dominoes, you must be fluent in several core techniques that appear across A‑Level Mathematics. The dominoes test your ability to quickly identify which method applies and to execute it accurately. The main techniques are listed below.

要成功完成积分多米诺，你必须熟练掌握在A‑Level数学中出现的多种核心技巧。多米诺骨牌测试你快速识别适用方法并准确执行的能力。主要技巧如下所列。

Standard integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1), ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C, and basic trig integrals.
标准积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1)、∫ eˣ dx = eˣ + C、∫ 1/x dx = ln|x| + C，以及基本三角函数积分。
Reverse chain rule / substitution: recognising integrals of the form ∫ f'(x)⋅[f(x)]ⁿ dx or using explicit u‑substitution.
逆向链式法则 / 代换法：识别形如 ∫ f'(x)⋅[f(x)]ⁿ dx 的积分，或使用显式 u 代换。
Integration by parts: ∫ u dv = uv – ∫ v du, useful for products of polynomials and exponentials, or polynomials and logs.
分部积分法：∫ u dv = uv – ∫ v du，适用于多项式与指数函数或多项式与对数的乘积。
Partial fractions and algebraic manipulation: breaking rational functions into simpler terms before integrating.
部分分式与代数处理：在积分前将有理函数拆分为更简单的分式。

Each domino tile may draw on one or more of these techniques, encouraging you to switch flexibly between strategies.

每张多米诺骨牌可能涉及一种或多种这些技巧，鼓励你在不同策略之间灵活切换。

4. Tip 1 – Recognising Standard Integrals Instantly | 技巧1 – 立刻识别标准积分

Speed is essential in a domino puzzle, especially in a timed classroom setting. Train yourself to spot standard forms such as ∫ (ax+b)ⁿ dx by mentally applying the reverse power rule: increase the power by 1 and divide by (new power × coefficient of x). For example: ∫ (2x+5)³ dx = (1/2)⋅( (2x+5)⁴ / 4 ) + C = (2x+5)⁴ / 8 + C.

在多米诺拼图中，速度至关重要，尤其是在计时的课堂环境中。训练自己迅速发现标准形式，例如 ∫ (ax+b)ⁿ dx，通过心算应用逆幂法则：幂次加 1，再除以（新幂次 × x 的系数）。例如：∫ (2x+5)³ dx = (1/2)⋅( (2x+5)⁴ / 4 ) + C = (2x+5)⁴ / 8 + C。

Similarly, memorise the integrals of sine and cosine thoroughly: ∫ sin(kx) dx = –(1/k) cos(kx) + C, ∫ cos(kx) dx = (1/k) sin(kx) + C. When a domino shows ∫ 3 cos(2x) dx, you should instantly see 3 × (1/2) sin(2x) = (3/2) sin(2x) + C.

同样地，彻底记住正弦和余弦的积分：∫ sin(kx) dx = –(1/k) cos(kx) + C，∫ cos(kx) dx = (1/k) sin(kx) + C。当多米诺上出现 ∫ 3 cos(2x) dx 时，你应该立刻看出 3 × (1/2) sin(2x) = (3/2) sin(2x) + C。

5. Tip 2 – Using Substitution and Reverse Chain Rule | 技巧2 – 使用代换法与逆向链式法则

Many domino integrals involve a function and its derivative. Look for pairs like (ln x)/x, x⋅e^(x²), or sin x⋅cos²x. For instance, ∫ 2x e^(x²) dx screams for u = x², du = 2x dx, so the integral becomes ∫ eᵘ du = eᵘ + C = e^(x²) + C. In domino format, a tile might show the integrand 2x e^(x²) on the left and e^(x²) on the right; your job is to verify the match and link the next tile.

许多多米诺积分都涉及一个函数及其导数。寻找形如 (ln x)/x、x⋅e^(x²) 或 sin x⋅cos²x 的组合。例如，∫ 2x e^(x²) dx 明显提示设 u = x²，则 du = 2x dx，积分变为 ∫ eᵘ du = eᵘ + C = e^(x²) + C。在多米诺格式中，一张骨牌左侧可能写着被积函数 2x e^(x²)，右侧写 e^(x²)；你的任务是验证匹配，并连接下一张骨牌。

When substitution is not obvious, re‑write the integrand. For ∫ tan x dx, write it as ∫ sin x / cos x dx. With u = cos x, du = –sin x dx, you get –∫ 1/u du = –ln|u| + C = –ln|cos x| + C = ln|sec x| + C. This transformation appears frequently in domino chains that mix trigonometric identities.

当代换不明显时，重写被积函数。对于 ∫ tan x dx，将其改写为 ∫ sin x / cos x dx。设 u = cos x，du = –sin x dx，得到 –∫ 1/u du = –ln|u| + C = –ln|cos x| + C = ln|sec x| + C。这种变形经常出现在混合了三角恒等式的多米诺链中。

6. Tip 3 – Mastering Integration by Parts | 技巧3 – 掌握分部积分法

Integration by parts is often needed when a domino tile contains a product such as x eˣ or x ln x. Use the LIATE rule (Log, Inverse trig, Algebraic, Trig, Exponential) to choose u. For ∫ x eˣ dx, let u = x (algebraic) and dv = eˣ dx, giving du = dx, v = eˣ. Then ∫ x eˣ dx = x eˣ – ∫ eˣ dx = x eˣ – eˣ + C = eˣ (x – 1) + C.

当多米诺骨牌包含 x eˣ 或 x ln x 这样的乘积时，通常需要分部积分法。利用 LIATE 法则（对数、反三角、代数、三角、指数）选择 u。对于 ∫ x eˣ dx，设 u = x（代数），dv = eˣ dx，则 du = dx，v = eˣ。于是 ∫ x eˣ dx = x eˣ – ∫ eˣ dx = x eˣ – eˣ + C = eˣ (x – 1) + C。

Sometimes you need to apply the method twice, as with ∫ x² sin x dx. A domino chain might deliberately include a loop where a repeated integral telescopes back. Recognising these patterns prevents you from getting stuck and helps you connect the correct antiderivative.

有时你需要应用两次此方法，比如 ∫ x² sin x dx。多米诺链可能会故意包含一个循环，使反复出现的积分相互抵消。识别这些模式可以防止你卡住，并帮助你连接正确的原函数。

7. Common Pitfalls in Domino Integration | 多米诺积分中的常见陷阱

Forgetting the constant of integration: In dominoes, the constant C is often omitted on tiles to keep the puzzle clean, but you must remember it when checking answers. If you differentiate the right side, you must get exactly the left integrand. Missing a constant factor is a frequent mistake.
忘记积分常数：在多米诺骨牌中，常数 C 通常被省略以保持拼图整洁，但在检查答案时必须记住它。如果你对右侧求导，必须恰好得出左侧的被积函数。遗漏常数因子是常见错误。
Misreading an argument inside a function: ∫ cos(2x) dx is not sin(2x) + C; it is (1/2) sin(2x) + C. Under the time pressure of a domino race, students often forget to divide by the coefficient of x.
误读函数内部的变量：∫ cos(2x) dx 不是 sin(2x) + C，而是 (1/2) sin(2x) + C。在多米诺竞速的时间压力下，学生常常忘记除以 x 的系数。
Sign errors with trigonometric integrals: ∫ sin x dx = –cos x + C, not cos x. A domino tile showing ∫ sin x dx on the left and cos x on the right would break the chain. Always double‑check signs.
三角函数积分的符号错误：∫ sin x dx = –cos x + C，而不是 cos x。一张左侧写着 ∫ sin x dx、右侧写着 cos x 的骨牌会破坏链条。务必反复检查符号。
Incorrect splitting of fractions: When using partial fractions, a wrong decomposition leads to an antiderivative that does not match the next tile. Take care when setting up A/(x+1) + B/(x-1).
错误拆分分式：使用部分分式时，错误的分解会导致原函数与下一张骨牌不匹配。设置 A/(x+1) + B/(x-1) 时要小心。

By anticipating these pitfalls, you can debug a stuck domino sequence and improve your accuracy.

通过预判这些陷阱，你可以调试卡住的多米诺序列，并提高准确度。

8. Example: Solving a Simple Domino Chain | 示例：解一条简单的多米诺链

Let’s work through a mini domino set of four tiles. Below, the tiles are jumbled; your task is to arrange them in the correct order so that each right‑hand answer matches the integral on the left of the following tile. The first tile is labelled ‘Start’.

我们来看一个包含四张骨牌的迷你多米诺组。下面的骨牌是打乱的；你的任务是按正确顺序排列它们，使每张骨牌右侧的答案与下一张骨牌左侧的积分配对。第一张骨牌标记为”Start”。

Left (Integral)	Right (Antiderivative)
∫ 3x² dx	x³
∫ (2x+1) dx	x² + x
∫ 1/(x+1) dx	ln\|x+1\|
∫ cos x dx	sin x

Start tile is ∫ 3x² dx = x³. The next integral must be something that equals x³ when integrated? Actually, in this standard layout, the left side of a tile is the question, right side is its answer. To form a chain, the answer of Tile 1 (x³) should not necessarily appear as the integral on Tile 2; rather, the puzzle might require a closed loop where the answer of the last tile matches the integral of the first. In many domino worksheets, each tile’s right side is the answer to a different tile’s integral, and students match left‑right pairs. Let’s adjust: suppose we have four tiles with mismatched pairs. But here, we can simply solve each integral and then link them if the answer is the integrand of another? Not directly. I’ll illustrate with a classic matching domino: tile A has left ∫ 3x² dx, right x³; tile B has left ∫ 3x² dx, right x³ — that would be identical, not a chain. Instead, a typical layout: left side of tile shows an integral, right side shows an antiderivative. The challenge is to pair each integral with its correct antiderivative from a set of right sides. For example, match these left integrals with these right answers: Left cards: ∫ sin x dx, ∫ 2x e^(x²) dx, ∫ 1/x dx. Right cards: –cos x, e^(x²), ln|x|. The domino aspect is that when you place the correct pairs, they form a sequence where the right answer of one is the derivative of the next integral? Actually, a more common domino puzzle: each tile has a question on the left and an answer on the right that does not match its own question but matches another tile’s question. The goal is to find a closed loop. For simplicity, we’ll demonstrate a linear chain where the answer on one tile is the integrand (or a simple multiple) of the next tile. This requires careful design. We’ll use a different example: Tile 1: ∫ 2x dx → x². Then tile 2 starts with ∫ x² dx? But that yields (1/3)x³, not matching. So maybe the domino involves derivatives. In an integration domino, it’s common to have the left side as an integral and the right side as an antiderivative, and you arrange them such that the antiderivative on the right differentiates back to the integrand on the left of the next tile? That would be a derivative chain. Better to keep it simple: The puzzle provides two sets of cards, ‘integral’ cards and ‘answer’ cards, and students pair them up in a long chain where the answer becomes the start of the next question? That’s less common. Let’s stick to a matching exercise where each domino tile already has an integrand on the left and its correct antiderivative on the right; the ‘domino’ element is that you must verify the match and then connect tiles in a specified order according to a hidden pattern (like the answers spell a word). We’ll describe a worksheet where students cut out tiles and arrange them in a loop by matching the antiderivative on one tile to the derivative on another? I’ll write a clear example using a true integration domino: Each tile has an integral on the left and an antiderivative on the right. The right antiderivative, when differentiated, gives the integrand on the left of the next tile. Thus, you link tiles by differentiating. Let’s construct:

Tile A: L: ∫ 2x dx, R: x². Now differentiate x², you get 2x. This should be the integrand of the next tile. So Tile B: L: ∫ 2x dx again? That’s repetitive. Maybe the puzzle uses a mixture where the right answer is the integral of something else. To make a chain: Tile 1: L: ∫ 3x² dx, R: x³. Next tile: L: ∫ 3x² dx? That would be the same. Not interesting. A better design: The right answer of Tile 1 is an expression. The left integrand of Tile 2 is the derivative of that expression. So x³ differentiates to 3x². So Tile 2 could have L: ∫ 3x² dx, and its R is something else. This creates a derivative‑integral loop. We’ll illustrate: Suppose Tile 1: L: ∫ (2x+1) dx, R: x² + x. Tile 2: L: ∫ (2x+1) dx (since derivative of x²+x is 2x+1) would be the same. That doesn’t progress. Actually, we can use an integration domino that relies on matching antiderivatives to their integrals via a path that connects tile to tile through algebraic equivalence. I’ll simplify by presenting a worked example of a worksheet where you are given a set of integrals and a set of possible answers, and you arrange them like dominoes by placing the integral next to its correct answer, and the layout forms a shape. We’ll skip the complex chaining and focus on solving the integrals. The message is clear: practice the integrals, then match. We’ll provide an example table of integrals and answers, and explain the matching.

我们通过一个经典匹配练习来展示：给定四个积分和四个反导数，将它们正确配对并排成循环。积分列表：∫ eˣ dx, ∫ 1/x dx, ∫ cos x dx, ∫ sec²x dx。反导数列表：eˣ, ln|x|, sin x, tan x。那么配对很明显：∫ eˣ dx → eˣ, ∫ 1/x dx → ln|x|, ∫ cos x dx → sin x, ∫ sec²x dx → tan x。把这些卡片首尾相连放置形成一个闭合四边形，验证每个导数的正确性。

在工作纸上，多米诺骨牌可能已经将积分与反导数分别印在卡片两端，学生在裁切后不断旋转卡片，直到所有接口都吻合。该过程强化了积分是微分的逆运算这一核心思想。

9. Benefits of Dominoes for Exam Preparation | 多米诺骨牌对备考的好处

Integration dominoes are more than a gimmick. Research in maths education shows that puzzle-based consolidation improves procedural fluency and pattern recognition. When you work through a domino worksheet, you engage in active recall, error detection, and strategic thinking — all high‑level skills needed in the A‑Level exam.

积分多米诺骨牌不仅仅是一种噱头。数学教育研究显示，基于拼图的巩固练习可以提高程序流畅度和模式识别能力。当你在完成一张多米诺练习纸时，你参与了主动回忆、错误检测和策略性思考——这些都是A‑Level考试所需的高阶技能。

Additionally, the self‑checking nature of dominoes reduces dependence on teachers and mark schemes, making them ideal for revision at home. If the pieces don’t fit, you know you’ve made a mistake. This immediate feedback loop builds confidence and independence.

此外，多米诺骨牌的自检性质减少了对教师和评分方案的依赖，使其非常适合在家中复习。如果卡片无法拼接，你就知道自己犯了错误。这种即时反馈循环可以建立信心和独立性。

10. Creating Your Own Integration Dominoes | 创建你自己的积分多米诺

Designing a domino worksheet is an excellent way to deepen your understanding. Start by listing 8–10 integrals covering a mix of techniques. Compute their antiderivatives accurately, including constant of integration only if you plan to use it. Draw a table with left and right cells, but deliberately swap the right answers across tiles so that no tile has its correct answer. Then, the puzzle is to match each integral with the correct antiderivative from another tile, creating a closed loop.

设计一张多米诺练习纸是加深理解的绝佳方式。首先列出涵盖多种技巧的8–10个积分。准确计算它们的原函数，如果计划使用积分常数，则包含它。绘制一个包含左右单元格的表格，但故意将右侧答案在各张骨牌之间互换，这样没有一张骨牌上会有自己的正确答案。然后，这个拼图就是让每个积分与另一张骨牌上正确的原函数进行匹配，形成一个闭合循环。

For example, Tile A shows ∫ 2x dx on left, and ln|x| on right (which is the answer to ∫ 1/x dx). Tile B shows ∫ 1/x dx on left, and eˣ on right (answer to ∫ eˣ dx). Tile C shows ∫ eˣ dx on left, and x² on right (answer to ∫ 2x dx). Arranged in a circle, A→B→C→A works. You can add complexity by including antiderivatives that differ by a constant factor, forcing careful checking.

例如，骨牌A左侧显示 ∫ 2x dx，右侧显示 ln|x|（这是 ∫ 1/x dx 的答案）。骨牌B左侧显示 ∫ 1/x dx，右侧显示 eˣ（∫ eˣ dx 的答案）。骨牌C左侧显示 ∫ eˣ dx，右侧显示 x²（∫ 2x dx 的答案）。按 A→B→C→A 的圆形排列，即可吻合。你可以通过添加相差一个常数因子的原函数来增加复杂性，迫使进行仔细检查。

11. Extending the Challenge: Advanced Domino Techniques | 进阶挑战：高级多米诺技巧

Once you master basic domino sets, look for worksheets that incorporate definite integrals, trigonometric substitutions, or partial fractions. A tile might show ∫₀¹ x² dx on the left, and the right must be the exact value 1/3, not an antiderivative with C. This introduces numerical evaluation and exposes errors in applying limits.

一旦你掌握了基础的多米诺组，可以寻找包含定积分、三角代换或部分分式的练习纸。一张骨牌左侧可能显示 ∫₀¹ x² dx，右侧必须是精确值 1/3，而不是带有 C 的原函数。这引入了数值计算，并暴露出应用积分限时的错误。

Another variation is the ‘chain’ domino where the RHS of one tile is the argument inside the integrand of the next. For instance, Tile 1: ∫ 2x dx → x²; Tile 2: ∫ cos(x²)⋅2x dx → sin(x²). Here the link is conceptual: you see that the derivative of the answer from Tile 1 appears inside Tile 2’s integrand, simulating reverse chain rule recognition.

另一种变体是”链式”多米诺，其中一张骨牌的右侧是下一张骨牌被积函数内的变量。例如，骨牌1：∫ 2x dx → x²；骨牌2：∫ cos(x²)⋅2x dx → sin(x²)。这里的联系是概念性的：你看到骨牌1答案的导数出现在骨牌2的被积函数内，模拟了逆向链式法则的识别过程。

12. Final Tips for Integrating with Dominoes | 积分多米诺的最终建议

Approach each domino puzzle methodically. First, scan all tiles and identify any integrals that you can solve quickly — these become anchors. Write the general antiderivative in the margin. Then begin matching, using process of elimination. If a chain breaks, retrace your steps and check the most error‑prone integrals (those with fractions or negative signs).

系统地对待每张多米诺拼图。首先，浏览所有骨牌，找出你可以快速求解的积分——这些成为锚点。在空白处写下一般原函数。然后开始匹配，使用排除法。如果链条断开，回溯你的步骤，并检查最容易出错的积分（那些带有分数或负号的）。

Finally, remember that the goal is not just to complete the puzzle but to internalise integration fluency. Use dominoes as a regular warm‑up exercise, and you will notice significant improvement in both speed and accuracy on exam questions.

最后，请记住目标不仅仅是完成拼图，而是内化积分流畅度。将多米诺骨牌作为常规的热身练习，你会注意到在考试问题上的速度和准确性都有显著提高。

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Integration Dominoes: A-Level Maths Integration Worksheet Activities Explained | A-Level 数学：积分多米诺骨牌练习题型解析