📚 Mixed Exercise 13: Integration | 混合练习13：积分
Welcome to this comprehensive revision guide for A-Level Edexcel Pure Mathematics, focusing on the concepts and problem types found in Mixed Exercise 13 – Integration. Whether you are preparing for an end‑of‑topic test or final AS/A‑Level examinations, mastering these integration techniques is essential. This article will walk you through the core principles of indefinite and definite integration, area calculations, identification of constants, and common pitfalls, all structured to mirror the mixed nature of the textbook exercise. We have paired every English explanation with a Chinese translation to support bilingual learners.
欢迎阅读这篇针对A-Level Edexcel纯数学的全面复习指南，重点梳理混合练习13——积分中的各种概念与题型。无论你是在准备章节测验还是AS/A2大考，掌握这些积分技巧都至关重要。本文将带你系统回顾不定积分、定积分、面积计算、积分常数确定以及常见错误，结构完全贴合教材混合练习的特点。每个英文段落都配有对应的中文解释，帮助双语学习者深入理解。
1. Integration Basics and the Power Rule | 积分基础与幂法则
tion (plus an arbitrary constant). The most fundamental rule is the power rule for integration: for any real number n ≠ -1, the integral of xⁿ with respect to x is given by ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. The symbol ∫ is the integral sign, dx indicates the variable of integration, and C is the constant of integration.
积分是微分的逆运算。如果我们已知一个函数的导数，通过积分就能还原出原函数（并加上一个任意常数）。最基本的法则是积分的幂法则：对于任意实数 n ≠ −1，xⁿ 对 x 的积分为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。符号 ∫ 是积分号，dx 表示积分变量，C 为积分常数。
pply the power rule term by term. Remember that constant terms also integrate: ∫ k dx = kx + C.
在爱德思混合练习13中，你经常会遇到需要积分如 3x²、5x⁴ 或 ½x⁻² 这样的式子。先把每一项写成 axⁿ 的形式，再逐项应用幂法则。记住常数项的积分是：∫ k dx = kx + C。
| Function (函数) | Integral (积分) |
|---|---|
| xⁿ (n ≠ -1) | xⁿ⁺¹/(n+1) + C |
| k (constant) | kx + C |
| 1/x (or x⁻¹) | ln|x| + C |
2. Finding the Constant of Integration | 确定积分常数
alue of C. After performing the indefinite integration, substitute the x and y coordinates of the given point into the equation and solve for C. This yields a particular solution rather than a general one.
如果题目给出了额外的信息，比如曲线 y = f(x) 经过的某个已知点，你就可以求出常数 C 的值。先求出不定积分，得到含有 C 的表达式，再将已知点的 x 和 y 坐标代入方程，解出 C。这样得到的就是特解而不是通解。
Mixed Exercise 13 frequently includes questions such as: 'The curve with equation y = f(x) passes through (2, 10) and f'(x) = 3x² + 2x. Find f(x).' Start by integrating f'(x) to get f(x) = x³ + x² + C, then use x = 2, y = 10 to find C.
混合练习13中经常出现这样的题目：“已知曲线 y = f(x) 过点 (2,10) 且 f'(x) = 3x² + 2x，求 f(x)。”先对 f'(x) 积分得 f(x) = x³ + x² + C，再利用 x=2，y=10 求出 C。
Always write the final answer with the constant explicitly substituted. Never leave C in the final equation unless the question asks for the general solution only.
最终答案一定要把常数的具体数值代入。除非题目只要求求通解，否则不要保留 C。
3. Definite Integration Fundamentals | 定积分基础
F(x)]ₐᵇ = F(b) − F(a), where F is any antiderivative of f. Notice that the constant C cancels out, so it is not needed in definite integral calculations.
定积分带有积分上下限，结果是一个数值。它通过微积分基本定理进行计算：∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)，其中 F 是 f 的任一个原函数。注意常数 C 在减法中会被抵消，因此计算定积分时不需要加上 C。
antiderivative x² + x, then substitute the limits: [(3)² + 3] − [(1)² + 1] = 12 − 2 = 10.
在混合练习13中，你需要计算类似 ∫₁³ (2x+1) dx 这样的定积分。先求原函数 x² + x，再代入上下限：[(3)²+3] − [(1)²+1] = 12 − 2 = 10。
Always show your substitution clearly, using square brackets and a clear subtraction step. Many marks are lost through careless arithmetic at the limit substitution stage.
务必清晰地展示代入步骤，使用方括号和明确的减法分步。很多学生因在代入上下限这一步的计算粗心而丢分。
4. Area Between a Curve and the x‑axis | 曲线与 x 轴之间的面积
When y = f(x) is non‑negative on the interval [a,b], the area enclosed between the curve, the x‑ A = ∫ₐᵇ f(x) dx. This is one of the most important geometric applications of integration.
当 y = f(x) 在区间 [a,b] 上非负时，由曲线、x 轴、直线 x=a 和 x=b 所围成的面积恰好等于定积分 A = ∫ₐᵇ f(x) dx。这是积分最重要的几何应用之一。
If, however, the curve dips below the x‑ here y ≥ 0 and y ≤ 0, integrate separately, and add the absolute values (or flip the sign for the negative parts).
然而，如果曲线的一部分位于 x 轴下方，由于高度为负，该部分的积分值会为负数。面积必须是正值，因此你需要将区间分割成 y≥0 和 y≤0 的几段，分别积分，再取绝对值相加（或将负值部分变号）。
e axis. Always sketch the graph or test the sign of y on each subinterval to avoid sign errors.
混合练习13特意包含了像 y = x² − 4 这样穿过 x 轴的曲线。一定要画草图或在子区间上检验 y 的正负，以免出现符号错误。
5. Area Between a Curve and the y‑axis | 曲线与 y 轴之间的面积
Sometimes you will be asked to find the area bounded by a curve and the y‑ and integrate with respect to y. The area is then A = ∫ₚ⁽ g(y) dy, where p and q are the y‑limits.
有时题目会要求你求出曲线与 y 轴围成的面积。这种情况下，将 x 表示为 y 的函数，即 x = g(y)，然后对 y 积分通常会更容易。面积公式为 A = ∫ₚ⁽ g(y) dy，其中 p 和 q 是 y 的上下限。
rtable with swapping variables is a key skill. For example, the area between the curve y = √x, the y‑ he lines y = 1 and y = 3 can be found by rewriting x = y² and integrating from y=1 to y=3.
在纯数学第一册的混合练习中，这类题目出现频率较低，但熟练掌握变量互换依然是一项关键技能。例如，求曲线 y = √x、y 轴以及直线 y=1 和 y=3 围成的面积时，可改写为 x = y²，然后对 y 从 1 到 3 积分。
Be meticulous with limits: the lower and upper limits now refer to y‑values, not x‑values. Label your diagram and switch your thinking clearly.
要特别注意积分上下限：此时上下限是 y 值，而不是 x 值。作图并清晰转换思维方式。
6. Area Between Two Curves | 两条曲线之间的面积
ₐᵇ [f(x) − g(x)] dx, provided f(x) ≥ g(x) throughout the interval. The integrand is simply 'top curve minus bottom curve'.
计算两条曲线 y = f(x) 与 y = g(x) 在 x=a 到 x=b 之间所围成的面积时，只要在该区间上恒有 f(x) ≥ g(x)，就可以使用公式 A = ∫ₐᵇ [f(x) − g(x)] dx。被积函数就是“上方曲线减去下方曲线”。
When the curves intersect multiple times, the 'top' and 'bottom' relationship may swap. You must find all intersection points, determine the relative position on each subinterval, and set up separate integrals. Mixed Exercise 13 challenges you with such piecewise area problems.
当两条曲线多次相交时，“上方”和“下方”的关系可能会互换。这时必须找出所有交点，在每个子区间上确定相对位置，并建立相应的分段积分。混合练习13会用这类分段面积问题来考验你。
Always subtract in the correct order. A common mistake is to integrate f(x) − g(x) when g(x) is actually the upper function, leading to a negative area that then requires sign correction. Sketching the graphs prevents this.
一定要按正确的顺序相减。常见的错误是在 g(x) 实为上方函数时却积分了 f(x)−g(x)，导致结果是负数，还得事后变号。绘制草图可以避免这种错误。
7. Handling Curves That Cross the Axis | 处理穿过坐标轴的曲线
When a curve crosses the x‑ within the integration interval, the definite integral alone does not give the total area – ke the absolute value of each portion (or write minus the integral for negative sections).
如果曲线在积分区间内穿过 x 轴，单纯计算定积分得到的是带有符号的净值面积，而非真正的几何面积。要获得真正的总面积，必须在 f(x)=0 的根处将积分拆开，并对每一部分取绝对值（或在负值部分前面添加负号）。
For example, the area between y = x³ − x and the x‑axis from x = −1 to x = 1 must be treated as ∫₋₁⁰ (x³−x) dx + |∫₀¹ (x³−x) dx| on [0,1] (or check by sign). This careful splitting is a typical mark scheme requirement.
例如，求 y = x³ − x 与 x 轴在 x=−1 到 x=1 之间的面积时，必须这样处理：∫₋₁⁰ (x³−x) dx + |∫₀¹ (x³−x) dx|，因为函数在 [−1,0] 上为正，在 [0,1] 上为负（或通过符号判断）。这种细致的分割是评分标准中的典型要求。
Never simply integrate across a root and take the absolute value of the whole result – that gives the wrong area if the curve oscillates. Always split and sum the absolute areas.
绝对不要直接跨根积分，然后对整个结果取绝对值——如果曲线上下波动，这样得到的面积是错误的。一定要分割并累加各个部分的绝对面积。
8. Reverse Differentiation and Checking Answers | 反微分验证答案
dx = F(x) + C, then differentiating F(x) must recover f(x). This quick check can catch 90% of algebraic mistakes in exams.
积分是反微分，因此你总可以通过对结果求导来验证不定积分是否正确。如果得到 ∫ f(x) dx = F(x) + C，那么对 F(x) 求导必须得到原函数 f(x)。这个快速检验能发现考试中90%的代数错误。
For definite integrals, you can approximate the area using a rough geometric estimate (e.g., counting grid squares on a sketch) to see if your numerical answer is sensible. If you expect a positive area of about 8 but your answer is −4.5, you have likely mixed up limits or signs.
对于定积分，你可以用粗略的几何估计（比如在草图上数格子）来看所得数值是否合理。如果你预期的正面积大约是8，结果却是−4.5，那很可能是上下限或符号弄反了。
In Mixed Exercise 13, time spent on verification is never wasted. The most reliable students routinely differentiate their integrated expressions to ensure they have applied the power rule correctly and not forgotten the constant multiplier.
在混合练习13中，花时间验证绝不是浪费。最可靠的学生都会习惯性地将积分结果求导，以确保正确应用了幂法则，且没有忘记系数。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
efinite integration problems. Even if the question does not immediately ask for C, writing '+ C' marks. Similarly, misapplying the power rule when n = −1 leads to an incorrect logarithm.
最常见的错误之一是在不定积分中忘记积分常数。即使题目没有明确要求标出 C，养成写“+ C”的习惯也是好的，爱德思考评通常会给方法分。类似地，当 n = −1 时误用幂法则会导致忘记使用对数。
et. If you integrate (2x+3)⁴, a common mistake is to write (2x+3)⁵/5 + C, forgetting the factor ½. The correct antiderivative is (1/2)*(1/5)*(2x+3)⁵ = (2x+3)⁵/10 + C.
另一个关键点是积分形如 (ax+b)ⁿ 的表达式时的符号与系数错误。记住要除以括号内 x 项的系数。若积分 (2x+3)⁴，常见错误是写成 (2x+3)⁵/5 + C，而忘记了因子 ½。正确的原函数是 (1/2)*(1/5)*(2x+3)⁵ = (2x+3)⁵/10 + C。
When computing areas, students sometimes assume the integral automatically gives area and take the final value as positive, without checking for roots. Always sketch, split, and justify your intervals. Marks are awarded for correct method as well as final answer.
在计算面积时，有些学生想当然地认为积分直接给出面积，于是直接把最终结果当作正值，而不检查是否有根。一定要画图、分段并说明区间的选取。评分不仅看最终答案，也看推理过程。
10. Worked Example from Mixed Exercise 13 | 混合练习13典型例题详解
Question: The curve C has equation y = f(x). It is given that f'(x) = 6x² − 2x + 1 and that C passes through the point (1, 4). Find f(x).
Solution: Integrate f' 1)³ − (1)² + 1 + C → 4 = 2 − 1 + 1 + C → 4 = 2 + C → C = 2. Hence f(x) = 2x³ − x² + x + 2.
题目： 曲线 C 的方程为 y = f(x)。已知 f'(x) = 6x² − 2x + 1，且曲线 C 经过点 (1, 4)。求 f(x)。
解答： 对 f'(x) 逐项积分：∫ (6x² − 2x + 1) dx = 2x³ − x² + x + C。故 f(x) = 2x³ − x² + x + C。代入点 (1,4)：4 = 2(1)³ − (1)² + 1 + C → 4 = 2 − 1 + 1 + C → 4 = 2 + C → C = 2。因此 f(x) = 2x³ − x² + x + 2。
xt, a typical area problem: Find the area enclosed between the curve y = x² − 2x and the x‑axis from x = 0 to x = 3.
First find where the curve crosses the x‑ 2. From 0 to 2 the curve is below the axis (negative), from 2 to 3 it is above. So area = |∫₀² (x²−2x) dx| + ∫₂³ (x²−2x) dx.
9) − (8/3 − 4) = (9 − 9) − (8/3 − 12/3) = 0 − (−4/3) = 4/3. Total area = 4/3 + 4/3 = 8/3.
再看一个典型的面积问题：求曲线 y = x² − 2x 与 x 轴在 x=0 到 x=3 之间围成的面积。
首先求曲线与 x 轴的交点：x²−2x=0 → x(x−2)=0 → x=0, 2。在 0 到 2 之间曲线在 x 轴下方（负值），在 2 到 3 之间在上方。因此面积 = |∫₀² (x²−2x) dx| + ∫₂³ (x²−2x) dx。
∫ (x²−2x) dx = (1/3)x³ − x²。计算：[0,2]：(8/3−4)−0 = 8/3−12/3 = −4/3，绝对值 4/3。[2,3]：(27/3−9)−(8/3−4) = (9−9)−(8/3−12/3) = 0−(−4/3)=4/3。总面积 = 4/3 + 4/3 = 8/3。
11. Summary and Key Formulae | 总结与关键公式
Let's consolidate the essential results that underpin every question in Mixed Exercise 13. Mastering these will give you the fluency needed to tackle the exercise confidently.
让我们巩固一下支撑混合练习13中每一道题的核心公式与结论。熟练掌握这些内容将使你游刃有余地应对所有题目。
- Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
- Constant multiple: ∫ k·f(x) dx = k ∫ f(x) dx
- Sum rule: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
- Special case: ∫ 1/x dx = ln|x| + C
- Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a)
- Area with x‑axis: A = ∫ₐᵇ |f(x)| dx (split at roots)
- Area between curves: A = ∫ₐᵇ (upper − lower) dx
中文对照：幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)； 常数倍：∫ k·f(x) dx = k ∫ f(x) dx； 和差法则：∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx； 特殊情况：∫ 1/x dx = ln|x| + C； 定积分：∫ₐᵇ f(x) dx = F(b)−F(a)； x 轴围成的面积：A = ∫ₐᵇ |f(x)| dx（在根处分段）； 曲线间面积：A = ∫ₐᵇ (上方−下方) dx。
When studying for Mixed Exercise 13, keep this formula list beside you. Write down every step, show the substitution of limits in definite integrals, and always include '+ C' in indefinite answers unless instructed otherwise.
在复习混合练习13时，请把这组公式放在手边。写下每一步推导，定积分要清晰展示上下限的代入过程，不定积分除非另有说明否则一律加上“+ C”。
12. Further Practice Suggestions | 进一步练习建议
To truly master the mixed exercise, you must go beyond passive reading. Begin by re‑attempting all the Mixed Exercise 13 questions from your textbook, timing each attempt. Then, challenge yourself with past paper questions on Integration from Edexcel AS and A‑level Mathematics – mixing up pure year 1 and some early year 2 integration for breadth.
要真正掌握这套混合练习，不能只停留在阅读层面。先把教材中混合练习13的所有题目重新限时做一遍。
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