📚 Edexcel A-Level Pure Maths: Integration Techniques and Applications | 爱德思A-Level纯数学：积分技巧与应用
Integration is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It appears in Pure Year 1 and is extended significantly in Year 2 with advanced techniques such as substitution, integration by parts, partial fractions and applications to areas, volumes and differential equations. This article collects the key methods, common traps and efficient exam strategies so that you can recognise the correct technique within seconds of reading a question.
积分是爱德思A-Level纯数学中考查最频繁的主题之一。它在纯数学第一年出现，在第二年大幅扩展，包括换元法、分部积分、部分分式，以及面积、体积和微分方程等应用。本文汇总关键方法、常见陷阱和高效应考策略，目标是让你在看到题目的几秒内就能判断出应该使用哪种技巧。
1. Standard Integrals and the Reverse Chain Rule | 标准积分与逆链式法则
Memorising the basic integrals is the foundation for every Edexcel integration question. The results must be automatic, especially the constant of integration when no limits are given. Marks are routinely lost for omitting + C, and examiners check it carefully in the first part of a longer question.
熟记基本积分公式是解答爱德思积分题的基础。这些结果必须能快速调用，不定积分尤其不要漏写积分常数。因漏掉 + C 而失分非常常见，在一道大题的第一部分，评分标准会专门检查这一项。
| f(x) | ∫ f(x) dx | 中文说明 |
|---|---|---|
| xⁿ, n ≠ −1 | xⁿ⁺¹ / (n + 1) + C | 幂函数积分 |
| eˣ | eˣ + C | 指数函数积分 |
| 1/x | ln|x| + C | 倒数函数积分 |
| sin x | −cos x + C | 正弦积分 |
| cos x | sin x + C | 余弦积分 |
| sec² x | tan x + C | 正割平方积分 |
Two linear extensions follow immediately from the table: sin(kx) integrates to −(1/k)cos(kx) + C, and cos(kx) integrates to (1/k)sin(kx) + C. Dividing by the coefficient k is the single most common slip in Year 1 questions.
由这张表可以立刻得到两个线性推广：sin(kx) 的积分是 −(1/k)cos(kx) + C，cos(kx) 的积分是 (1/k)sin(kx) + C。忘记除以系数 k 是第一年题目中最常见的失误。
The reverse chain rule is essentially integration by recognising a derivative product. If you see a composite function multiplied by the derivative of its inner function, you can write down the answer directly without a formal substitution.
逆链式法则本质上是通过识别导数乘积来积分。如果你看到复合函数乘以其内层函数的导数，就可以不经正式换元而直接写出结果。
∫ f'(g(x)) g'(x) dx = f(g(x)) + C
A closely related pattern is ∫ f'(x) / f(x) dx = ln|f(x)| + C, which covers cases like ∫ 2x / (x² + 5) dx = ln|x² + 5| numerator that is exactly the derivative of the denominator, or a constant multiple of it.
一个紧密相关的模式是 ∫ f'(x) / f(x) dx = ln|f(x)| + C，例如 ∫ 2x / (x² + 5) dx = ln|x² + 5| + C。考试时要留意分子是否恰好是分母的导数，或者只差一个常数倍。
The factor of 2 is what turns 3x² into 6x², so the constant outside the sine is exactly 2.
例子：∫ 6x² cos(x³) dx = 2 sin(x³) + C。因为 x³ 的导数是 3x²，而 6x² = 2 × 3x²。这个倍数 2 正是把 3x² 变成 6x² 的关键，所以正弦前面的系数恰好是 2。
2. Definite Integrals and Areas | 定积分与面积
A definite integral is evaluated by finding an antiderivative first and then substituting the limits. The constant of integration cancels, so it is normally omitted, but the square-bracket notation must be clear to secure the method mark.
定积分的做法是先求原函数，再代入上下限。积分常数会相互抵消，因此通常省略不写，但方括号写法必须规范，才能拿到方法分。
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)
The area between a curve and the x-axis is found by integrating, but the integral measures signed area. Where the curve dips below the axis, the integral is negative, so you must split the region at the roots and add the absolute values of each part.
曲线与 x 轴之间的面积可以通过积分求得，但积分给出的是有向面积。当曲线落到 x 轴下方时积分为负，因此必须在与 x 轴的交点处把区域切开，再取各部分绝对值相加。
For the area between two curves, subtract the lower function from the upper one and integrate between the intersection points. Sketching first is not optional: it prevents you from swapping the order and losing all the accuracy marks.
求两条曲线之间的面积时，用上方函数减去下方函数，在交点之间积分。先画草图不是可选项：它能避免上下颠倒，从而丢掉全部精度分。
x − 2 = 0, so x = −1 and x = 2. The area is ∫₋₁² (x + 2 − x²) dx = 9/2 = 4.5 square units.
例题：求 y = x² 与 y = x + 2 之间的面积。两曲线相交处满足 x² − x − 2 = 0，即 x = −1 与 x = 2。面积为 ∫₋₁² (x + 2 − x²) dx = 9/2 = 4.5 个平方单位。
3. Integration by Substitution | 换元积分法
Substitution is the systematic version of the reverse chain rule and is the standard tool when a composite function has an awkward inner expression. The rule in the formula book is written as follows.
换元法是逆链式法则的系统化版本，当复合函数的内层表达式较为复杂时，它就是标准工具。公式表中的换元公式写作如下形式。
∫ f(x) dx = ∫ f(u) (dx/du) du
The procedure has four fixed steps. Choose u, differentiate to get du/dx, rewrite every x and dx in terms of u, integrate, then convert back to x unless the limits were changed. Missing the dx term is the classic error and costs both accuracy marks in a two-mark conversion.
换元法有四个固定步骤：选取 u；求导得到 du/dx；把所有的 x 与 dx 都改写成 u 的表达式；积分后再换回 x，除非你已经同步更换了积分限。漏写 dx 是经典错误，会在两分的改写环节中全部丢分。
For a definite integral, change the limits at the same time as you change the variable. This avoids the long and error-prone process of substituting back before evaluating.
对于定积分，要在换元的同时更换积分限。这样可以避免先把变量换回去再代入，省去冗长且容易出错的过程。
when x = 1, u = 2. The integral becomes ∫₁² u⁵ du = [u⁶ / 6]₁² = (64 − 1)/6 = 21/2 = 10.5.
例题：计算 ∫₀¹ 2x(x² + 1)⁵ dx。令 u = x² + 1，则 du = 2x dx。当 x = 0 时 u = 1；当 x = 1 时 u = 2。原式化为 ∫₁² u⁵ du = [u⁶ / 6]₁² = (64 − 1)/6 = 21/2 = 10.5。
tutions worth recognising instantly include u = x² + 1 for polynomials inside a power, u = √x 1 for algebraic fractions, and u = ln x when a logarithmic term appears in the numerator.
值得一眼识别的常见换元包括：幂内多项式取 u = x² + 1；代数分式取 u = √x 或 u = x + 1；当分子出现对数项时取 u = ln x。
4. Integration by Parts | 分部积分
Integration by parts is the reverse of the product rule and is the required technique whenever two unrelated families of functions are multiplied together, such as x with eˣ, or ln x with 1.
分部积分是乘积法则的逆运算，只要两个不同类函数相乘，例如 x 与 eˣ，或 ln x 与 1，就必须使用它。
∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx
Choosing u correctly saves time. The standard priority order is logarithmic, algebraic, trigonometric, exponential, which is often remembered as the order in which you should let u appear. If a term is missing, treat it as multiplied by 1.
正确选取 u 能节省时间。标准优先顺序是：对数函数、代数函数、三角函数、指数函数，这也是决定谁来做 u 的顺序。如果缺少某一项，就把它看作乘以 1。
/dx = 1, so du/dx = 1/x and v = x. The result is x ln x − ∫ x × (1/x) dx = x ln x − x + C.
例题：∫ ln x dx。取 u = ln x，dv/dx = 1，则 du/dx = 1/x，v = x。结果为 x ln x − ∫ x × (1/x) dx = x ln x − x + C。
Some integrals require the technique twice. For ∫ x² eˣ dx, the first application reduces x² to x, and the second reduces x to a constant. For ∫ eˣ returns the original integral, which you then move to the left-hand side and divide by 2.
有些积分需要连续使用两次。对于 ∫ x² eˣ dx，第一次把 x² 降到 x，第二次把 x 降为常数。对于 ∫ eˣ sin x dx，两次分部积分后会重新出现原积分，此时把它移到等式左边再除以 2 即可。
∫ eˣ sin x dx = (eˣ (sin x − cos x)) / 2 + C
For a definite integral by parts, apply the limits to the uv term first and then handle the remaining integral. Mixing the two stages is the most frequent source of arithmetic errors in this topic.
定积分使用分部积分时，应先对 uv 项代入上下限，再处理剩下的积分。把两个阶段的上下限混在一起，是本主题最常见的计算错误来源。
5. Integration with Partial Fractions | 部分分式与积分
Rational functions with factorisable denominators should be split into partial fractions before integrating. The resulting terms are logarithms or simple powers, both of which are directly integrable.
分母可因式分解的有理函数，应先拆成部分分式再积分。拆开后得到的是对数项或简单幂函数，都能直接积分。
For distinct linear factors, write the fraction with unknown constants A and B, multiply through by the denominator and substitute convenient values of x to find them. Substituting the root of each factor is faster than comparing coefficients.
对于互不相同的线性因式，先写出含未知常数 A 与 B 的表达式，两边乘以分母，再代入方便的 x 值求出它们。代入每个因式的根比比较系数更快。
(x + 1)(x + 2)) dx. Since 1 / ((x + 1)(x + 2)) = 1/(x + 1) − 1/(x + 2), the integral is ln|x + 1| − ln|x + 2| + C, which may also be written as ln|(x + 1)/(x + 2)| + C.
例题：∫ 1 / ((x + 1)(x + 2)) dx。因为 1 / ((x + 1)(x + 2)) = 1/(x + 1) − 1/(x + 2)，所以积分结果为 ln|x + 1| − ln|x + 2| + C，也可写成 ln|(x + 1)/(x + 2)| + C。
Three special cases need extra care. Repeated factors require a term for every power, such as A/(x + 1) + B/(x + 1)². An improper fraction must first be divided so the numerator degree is lower than the denominator degree before splitting. A quadratic factor that does not factorise produces an arctangent term of the form (1/a) tan⁻¹(x/a) + C.
三种特殊情况需要额外留意：重复因式要为每一个幂次各设一项，例如 A/(x + 1) + B/(x + 1)²；假分式必须先用除法降次，使分子次数低于分母次数，然后再拆分；无法分解的二次因式会产生反正切项，形式为 (1/a) tan⁻¹(x/a) + C。
6. Trigonometric Identities and Integrals | 三角恒等式与三角积分
Powers of trigonometric functions are handled by rewriting them with double-angle identities. This converts an integral you cannot do into one you can, and the transformation is almost always worth a method mark on its own.
三角函数的高次幂要借助二倍角恒等式改写。这样可以把做不出的积分转化成能做的积分，而这一步改写本身通常就能拿到方法分。
sin² x = ½ (1 − cos 2x) cos² x = ½ (1 + cos 2x)
2x)/4 + C. Notice that the x/2 term comes from integrating the constant ½, and students frequently forget it.
应用后得到 ∫ sin² x dx = x/2 − sin(2x)/4 + C 与 ∫ cos² x dx = x/2 + sin(2x)/4 + C。注意 x/2 这一项来自常数 ½ 的积分，学生常常漏掉它。
= sec² x − 1, so ∫ tan² x dx = tan x − x + C. The related standard result ∫ tan x dx = ln|sec x| + C should also be memorised, since it appears in differential equation questions.
对于 tan² x，利用 tan² x = sec² x − 1，可得 ∫ tan² x dx = tan x − x + C。与之相关的标准结果 ∫ tan x dx = ln|sec x| + C 也应记住，因为它常出现在微分方程题目中。
If you see a single odd power of sine or cosine, do not use a double-angle identity; strip one factor and substitute instead.
诸如 sin 3x cos 2x 的乘积要用积化和差公式处理，而 sin 2x = 2 sin x cos x 用于处理奇次幂。如果遇到正弦或余弦的奇次幂，不要使用二倍角恒等式，而应拆出一个因子后用换元法。
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