📚 Mixed Exercise 13: Integration | 混合练习13：积分
Mixed Exercise 13 is the end-of-chapter mixed review for Chapter 13 of the Edexcel AS Pure Mathematics Year 1 course. It brings together indefinite integration, definite integration, finding the constant of integration, and area problems under curves and between curves.
混合练习13是Edexcel AS纯数学一年级第13章的章末综合练习。它综合考查不定积分、定积分、积分常数的求解，以及曲线下方和曲线之间的面积问题。
1. Overview of Mixed Exercise 13 | 混合练习13 总览
This exercise is designed to test every skill from the integration chapter in one mixed set. You will see polynomials, powers of x, definite integrals with numerical limits, curves defined through their derivatives, and area problems that require you to choose the correct limits and subtract the correct functions.
本练习旨在用一组混合题考查积分章节的全部技能。你会遇到多项式、x 的幂、带数值上下限的定积分、由导数定义的曲线，以及需要你正确选择上下限并正确相减的面积问题。
In Edexcel AS Mathematics, integration is treated as the reverse process of differentiation. Mastery of the basic rule for xⁿ and careful use of the constant of integration are the two most important foundations.
在Edexcel AS数学中，积分被看作微分的逆运算。熟练掌握 xⁿ 的基本积分公式，并正确使用积分常数，是两个最重要的基础。
The mixed exercise is deliberately unordered. A question on a definite integral may be followed by one on a curve defined by dy/dx, and then by an area question involving two curves. Your job is to recognise quickly which technique each part needs before you start writing.
混合练习的题目顺序是刻意打乱的。一道定积分题之后可能紧跟一道由 dy/dx 定义的曲线题，再之后是涉及两条曲线的面积题。你的任务是动笔之前就迅速判断每一小问需要哪一种方法。
2. The Integration Toolkit | 积分工具包
rovided n ≠ -1. The constant c appears because differentiation of any constant gives zero.
本练习的核心是微分的逆运算。如果 dy/dx = xⁿ，那么 y = xⁿ⁺¹/(n + 1) + c，前提是 n ≠ -1。常数 c 出现的原因是任何常数求导后都为零。
| Function f(x) | Indefinite integral ∫ f(x) dx |
|---|---|
| xⁿ, n ≠ -1 | xⁿ⁺¹/(n + 1) + c |
| k, where k is a constant | kx + c |
| a xⁿ, a is a constant | a xⁿ⁺¹/(n + 1) + c |
| 1 | x + c |
| x | x²/2 + c |
| x² | x³/3 + c |
Constants are integrated as kx, and sums or differences are integrated term by term. Never try to integrate a product by integrating each factor separately unless it has been expanded into a sum first.
常数项积分后为 kx，和式或差式可逐项积分。除非先把乘积展开为和式，否则不要通过分别积分每个因式来积分乘积。
¹/(n + 1) would give division by zero. The integral of x⁻¹, which involves the natural logarithm, is covered later in the A-level course and does not appear in this exercise.
n = -1 的情况被排除，因为 xⁿ⁺¹/(n + 1) 会造成除以零。x⁻¹ 的积分涉及自然对数，会在 A-level 后续课程中学习，不出现在本练习中。
3. Indefinite Integration and the Constant c | 不定积分与积分常数
An indefinite integral has no limits and must always end with "+ c". A common mark-loss in this exercise is forgetting the constant, especially when the question does not explicitly ask for it.
不定积分没有上下限，结果末尾必须写上 "+ c"。本练习中最常见的失分点就是忘记写常数，尤其是在题目没有明确要求的时候。
ed separately: 6x² gives 6x³/3 = 2x³, 4x gives 4x²/2 = 2x², and the constant −5 gives −5x.
例如：∫ (6x² + 4x − 5) dx = 2x³ + 2x² − 5x + c。每一项单独处理：6x² 得到 6x³/3 = 2x³，4x 得到 4x²/2 = 2x²，常数 −5 得到 −5x。
∫ (a xⁿ + b xᵐ + c) dx = a xⁿ⁺¹/(n + 1) + b xᵐ⁺¹/(m + 1) + c x + C
Check every answer by differentiating it. If the derivative does not return the original integrand, something has gone wrong. This one habit catches almost every arithmetic slip in the exercise.
把每个答案求导检验一遍。如果导数不能还原原来的被积函数，说明出了错。养成这个习惯，几乎可以查出练习中所有的运算失误。
4. Finding the Constant from a Given Point | 由已知点求积分常数
When a curve is defined by its gradient function dy/dx and you are given a point on the curve, you integrate first and then substitute the point to find c. This is the "particular solution" type of question.
当曲线由其梯度函数 dy/dx 定义，并给出曲线上一点时，你先积分，再代入该点求出 c。这就是"特解"型题目。
4x² + 5x + c. Substituting x = 2, y = 9 gives 8 − 16 + 10 + c = 9, so 2 + c = 9 and c = 7.
例题：dy/dx = 3x² − 8x + 5，且曲线经过点 (2, 9)。积分得 y = x³ − 4x² + 5x + c。代入 x = 2，y = 9 得 8 − 16 + 10 + c = 9，即 2 + c = 9，所以 c = 7。
y = x³ − 4x² + 5x + 7
Always write the general form with "+ c" first, then do the substitution on a fresh line. Trying to find c mentally in one step is where most sign errors occur, particularly when the given point has a negative coordinate.
务必先写出带 "+ c" 的一般形式，再另起一行代入求值。试图一步心算求出 c，是符号错误最多的地方，尤其是当已知点含有负坐标时。
nt on y has dy/dx = 0 there, so you can also use it to check the gradient function itself.
如果题目给出的是驻点而不是一般点，要记住 y 上的驻点处 dy/dx = 0，因此也可以用它来检验梯度函数本身是否写对。
5. Definite Integration and Limits | 定积分与上下限
A definite integral has numerical limits and produces a number, not a function. The constant c cancels automatically, so it is not written after the square bracket.
定积分有数值上下限，结果是一个数而不是函数。常数 c 会自动抵消，因此方括号后面不写 c。
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)
The standard layout is: integrate inside square brackets, then write the limits on the bracket, then substitute the upper limit minus the lower limit. Writing the bracket clearly is worth marks even if your arithmetic later slips.
标准写法是：先在方括号内积分，再在括号上标出上下限，然后代入上限值减下限值。即使后面算错，把方括号写清楚也能拿到步骤分。
Example: ∫₁³ (x² + 2) dx = [x³/3 + 2x]₁³ = (9 + 6) − (1/3 + 2) = 15 − 7/3 = 38/3.
例如：∫₁³ (x² + 2) dx = [x³/3 + 2x]₁³ = (9 + 6) − (1/3 + 2) = 15 − 7/3 = 38/3。
Take care with negative limits: substituting a negative lower limit means every odd power changes sign. Bracketing each substitution in its own set of brackets, as above, is the safest habit.
注意负的上下限：代入负的下限时，每一次奇数次幂都会变号。像上面那样把每次代入单独用一对括号括起来，是最稳妥的习惯。
6. Area Under a Curve | 曲线下方的面积
ite integral of f(x) from a to b, provided the curve lies above the axis on that interval.
曲线 y = f(x)、x 轴以及竖直线 x = a 和 x = b 之间的面积，等于 f(x) 从 a 到 b 的定积分，前提是曲线在该区间上位于 x 轴上方。
Area = ∫ₐᵇ y dx
between x = 0 and x = 4. Then ∫₀⁴ (4x − x²) dx = [2x² − x³/3]₀⁴ = (32 − 64/3) − 0 = 32/3.
例题：求 y = 4x − x² 在 x = 0 与 x = 4 之间曲线下方的面积。则 ∫₀⁴ (4x − x²) dx = [2x² − x³/3]₀⁴ = (32 − 64/3) − 0 = 32/3。
If part of the curve lies below the x-axis, the definite integral over that part is negative. To get a geometric area you must split the interval at the point where the curve crosses the axis and take the modulus of each piece before adding.
如果曲线有一部分位于 x 轴下方，该部分上的定积分是负值。要得到几何面积，必须在曲线与 x 轴交点处把区间拆开，对每一段取绝对值后再相加。
Sketching the curve first is not optional in an exam. A quick sketch shows you where the roots are, whether the curve is above or below the axis, and whether your final answer should be a sensible size.
考试中先画草图不是可选项。快速草图能显示根的位置、曲线在 x 轴上方还是下方，以及最终答案是否在合理范围内。
7. Area Between Two Curves | 两条曲线之间的面积
lose a region, the area is the integral of the upper curve minus the lower curve between the points where they meet.
当两条曲线 y = f(x) 与 y = g(x) 围成一个区域时，面积等于在两者交点之间对"上方曲线减下方曲线"求积分。
Area = ∫ₐᵇ (yupper − ylower) dx
2x is above the parabola, so the area is ∫₀² (2x − x²) dx = [x² − x³/3]₀² = 4 − 8/3 = 4/3.
例题：曲线 y = x² 与 y = 2x 相交于 x² = 2x，即 x = 0 与 x = 2。在这两个值之间，直线 y = 2x 在抛物线之上，所以面积是 ∫₀² (2x − x²) dx = [x² − x³/3]₀² = 4 − 8/3 = 4/3。
Subtracting the wrong way round gives a negative answer. If your result comes out negative, you have almost certainly put the lower curve first, so change the sign rather than starting the whole calculation again.
相减方向弄反会得到负值。如果结果算出负数，几乎可以肯定你把下方曲线写在了前面，直接改变符号即可，不必整道题重算。
An alternative method is to integrate each curve separately and subtract the two areas, but the single integral of the difference is usually faster and less prone to rounding errors.
另一种方法是分别对两条曲线积分再相减，但对差值作单次积分通常更快，也更不容易产生舍入误差。
8. Choosing the Correct Limits | 正确选择积分上下限
In area questions the limits are often not given directly. You have to find them from the information in the question, and choosing them wrongly is the single biggest source of lost marks in this exercise.
在面积题中，上下限往往不会直接给出。你必须从题目信息中求出它们，而选错上下限是本练习中失分最多的原因。
- Where a curve meets the x-axis, set y = 0 and solve for x.
- Where two curves meet, set the two expressions equal and solve.
- Where a line x = a is given, that value is one of your limits.
- Where the region is bounded by a curve and a line, find the intersection before integrating.
- 曲线与 x 轴相交处，令 y = 0 解出 x。
- 两条曲线相交处，令两式相等并求解。
- 若给出直线 x = a，该值就是上下限之一。
- 若区域由曲线与直线围成，先求交点再积分。
Always write down the limits you have found before you integrate. Naming them a and b, and stating which is upper and which is lower, makes the later substitution mechanical and protects your method marks.
积分之前先把找到的上下限写下来。把它们记为 a 和 b，并说明哪个是上限、哪个是下限，能让后面的代入变成机械操作，保护你的方法分。
9. Expanding Before Integrating | 先展开再积分
You cannot integrate a product or a quotient by treating the factors separately. If you see something like x(x + 3), expand it to x² + 3x first, then integrate term by term.
不能把乘积或商的各个因式分开积分。遇到 x(x + 3) 这样的式子，先展开成 x² + 3x，再逐项积分。
/3 + 3x²/2 + c. Integrating x and (x + 3) separately and multiplying the results would be completely wrong.
例如：∫ x(x + 3) dx = ∫ (x² + 3x) dx = x³/3 + 3x²/2 + c。把 x 与 (x + 3) 分别积分再相乘是完全错误的做法。
The same applies to brackets such as (x + 1)(x − 4) or (2x − 1)², and to expressions written as fractions with a single term in the denominator, which can be split into separate powers of x.
同样适用于 (x + 1)(x − 4) 或 (2x − 1)² 这类括号，也适用于分母只有单项的分式，它们可以拆成若干个 x 的幂。
Rewrite roots and reciprocals as powers before integrating: √x comes x^½ and 1/x² becomes x⁻². The power rule then applies directly with n = ½ or n = −2.
积分前先把根式和倒数改写成幂的形式：√x 写成 x^½，1/x² 写成 x⁻²。之后幂法则就能直接使用，n = ½ 或 n = −2。
10. Common Errors and Exam Technique | 常见错误与考试技巧
Most marks lost in this exercise come from a small number of repeated slips rather than from genuine misunderstanding of the mathematics.
本练习中丢掉的分数，大多来自少数几个反复出现的失误，而不是对数学本身的真正误解。
- Forgetting "+ c" in an indefinite integral.
- Dividing by n instead of by n + 1.
- Adding the wrong power, for example turning x³ into x⁴/4 by mistake with the wrong index.
- Substituting the limits in the wrong order, that is F(a) − F(b) instead of F(b) − F(a).
- Treating a product as if it could be integrated factor by factor.
- Giving a negative area without interpreting it, or without checking the curve position.
- 不定积分忘记写 "+ c"。
- 除以 n 而不是除以 n + 1。
- 幂次加错，例如把 x³ 错误地写成 x⁴/4。
- 上下限代入顺序颠倒，即写成 F(a) − F(b) 而不是 F(b) − F(a)。
- 把乘积当作可以逐个因式积分。
- 算出负面积却不作解释，也不检查曲线位置。
Show every step. In this exercise method marks are generous: writing the bracket, stating the limits, and showing the substitution can earn most of the credit even if a final arithmetic error occurs.
把每一步都写出来。本练习的方法分给得很宽：写出方括号、标明上下限、展示代入过程，即使最后算错，也能拿到大部分分数。
11. Worked Exam-Style Examples | 典型考题演练
ves y = 2x³ − x² + c, and substituting gives 2 − 1 + c = 4, so c = 3 and y = 2x³ − x² + 3.
例题A：已知 dy/dx = 6x² − 2x，曲线过点 (1, 4)，求 y。积分得 y = 2x³ − x² + c，代入得 2 − 1 + c = 4，所以 c = 3，y = 2x³ − x² + 3。
Example B: evaluate ∫₀² (3x² + 4x) dx. The bracket gives [x³ + 2x²]₀² = (8 + 8) − 0 = 16.
例题B：计算 ∫₀² (3x² + 4x) dx。方括号给出 [x³ + 2x²]₀² = (8 + 8) − 0 = 16。
+ 2 − x²) dx = [x²/2 + 2x − x³/3]₋₁² = (2 + 4 − 8/3) − (1/2 − 2 + 1/3) = 10/3 + 7/6 = 9/2.
例题C：求 y = x² 与 y = x + 2 之间的面积。两者相交于 x² − x − 2 = 0，即 x = −1 与 x = 2。其间直线在抛物线之上，故 + 2 − x²) dx = [x²/2 + 2x − x³/3]₋₁² = (2 + 4 − 8/3) − (1/2 − 2 + 1/3) = 10/3 + 7/6 = 9/2。
Notice how each of the three examples uses a different sub-skill: finding a constant, evaluating a definite integral, and setting up an area between two curves. The mixed exercise is built exactly this way.
注意这三道例题分别用了不同的子技能：求常数、计算定积分、以及建立两曲线间面积的表达式。混合练习正是按这种方式编排的。
12. Revision Checklist for Mixed Exercise 13 | 混合练习13 复习清单
Before you move on to the next chapter, work through this checklist and make sure every statement is true for you.
在进入下一章之前，逐条核对下面的清单，确保每一条对你来说都成立。
- I can integrate xⁿ, constants, and multiples of powers of x.
- I always include "+ c" for indefinite integrals and can explain why.
- I can find c from a point on the curve, and from a stationary point.
- I can evaluate a definite integral using the correct order F(b) − F(a).
- I can find an area under a curve and split the interval when the curve crosses the axis.
- I can find the area between two curves after solving for their intersection points.
- I expand products and rewrite roots as powers before integrating.
- I check my answers by differentiating, where possible.
- 我能对 xⁿ、常数以及 x 的幂的倍数进行积分。
- 我在不定积分中总会写 "+ c"，并能解释原因。
- 我能由曲线上一点、也能由驻点求出 c。
- 我能按正确的顺序 F(b) − F(a) 计算定积分。
- 我能求曲线下方的面积，并在曲线穿过 x 轴时拆分区间。
- 我能在解出交点后求两条曲线之间的面积。
- 我会在积分前展开乘积，并把根式改写为幂。
- 我会尽可能通过求导来检验答案。
Mixed Exercise 13 rewards accuracy more than cleverness. Every question in it is built from the same handful of rules, so the goal is not to learn new theory but to apply the existing rules reliably under exam conditions.
混合练习13更看重准确而非技巧。其中每一道题都由同一套为数不多的规则搭建而成，所以目标不是学新理论，而是在考试条件下可靠地运用已有规则。
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