📚 Edexcel A-Level Maths: Integration Techniques | 爱德思 A-Level 数学：积分技巧
Integration is one of the highest-weighted topics in the Edexcel A-Level Mathematics Pure specification. This article brings together the essential techniques: reversing differentiation, substitution, integration by parts, partial fractions, parametric integration, separable differential equations, area and volume calculations, and numerical estimation. It is written to help you work fluently under exam conditions and avoid losing accuracy marks.
积分是爱德思 A-Level 数学 Pure 部分权重很高的主题。本文汇总核心方法：逆向微分、换元积分、分部积分、部分分式积分、参数方程积分、可分离变量微分方程、面积与旋转体体积计算以及数值估计，帮助你适应考试节奏并减少细节失分。
1. Integration as Reverse Differentiation | 积分作为逆微分
At A-Level, the indefinite integral of a function f'(x) is written as ∫ f' C. The constant C is essential: Edexcel mark schemes routinely deduct the final A mark if '+ C' is missing on an indefinite integral.
在 A-Level 中，f'(x) 的不定积分写作 ∫ f'(x) dx = f(x) + C。常数 C 必不可少，Edexcel 评分方案通常会在不定积分缺少 '+ C' 时扣掉最后一个 A 分。
This means every time you reverse a derivative, you must add the arbitrary constant unless the question already gives a boundary condition. A definite integral does not need '+ C' because the constant cancels when you evaluate F(b) - F(a).
这意味着每次你逆向求导时，都必须加上任意常数，除非题目已经给出边界条件。定积分不需要 '+ C'，因为在计算 F(b) - F(a) 时常数会抵消。
∫ f'(x) dx = f(x) + C
2. Standard Results and Linearity | 标准积分结果与线性性质
Edexcel expects you to know a core set of standard integrals without a formula booklet. You should recognise powers, exponentials, reciprocal functions, trigonometric functions, and the inverse tangent result immediately.
爱德思要求你无需公式册就能掌握一组核心标准积分。你应当立即识别幂函数、指数函数、倒数函数、三角函数以及反正切结果。
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ -1
∫ 1/x dx = ln |x| + C
∫ eˣ dx = eˣ + C
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ sec² x dx = tan x + C
∫ 1/(1 + x²) dx = arctan x + C
Integration is linear, so you can split sums and take constants outside the integral sign. This rule lets you integrate polynomial expressions term by term and handle coefficients without rewriting the whole expression.
积分具有线性性质，因此你可以拆分和式并把常数移到积分号外。这条规则让你逐项积分多项式表达式，并处理系数而不必重写整个式子。
∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx
3. Definite Integrals and the Fundamental Theorem | 定积分与微积分基本定理
d area. The fundamental theorem of calculus connects differentiation and integration: if F'(x) = f(x), then the definite integral from a to b equals F(b) - F(a).
定积分带有上下限，表示有向面积。微积分基本定理把微分和积分联系起来：若 F'(x) = f(x)，则从 a 到 b 的定积分等于 F(b) - F(a)。
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) - F(a)
Always substitute the upper limit first and subtract the lower limit result. Use brackets to avoid sign errors, especially when the antiderivative contains negative terms or trigonometric functions.
务必先代入上限再减去下限结果。使用括号以避免符号错误，尤其当原函数含有负项或三角函数时。
If the curve lies below the x-axis, the definite integral is negative. If a question asks for total area, you must split the integral at the roots and take absolute values of each part.
如果曲线位于 x 轴下方，定积分为负。如果题目要求总面积，你必须在根处拆分积分，并对每一部分取绝对值。
4. Integration by Substitution | 换元积分法
Substitution reverses the chain rule. Choose u = g(x), then differentiate to get du/dx = g'(x), so du = g'(x) dx. Rewrite the entire integral in terms of u and du before integrating.
换元积分法是链式法则的逆运算。令 u = g(x)，求导得 du/dx = g'(x)，所以 du = g'(x) dx。在积分前，把整个积分改写成关于 u 和 du 的形式。
∫ f(g(x)) g'(x) dx = ∫ f(u) du
For definite integrals, you must change the limits from x-values to u-values. Alternatively, integrate in u, substitute back to x, and then apply the original limits. Changing limits is usually faster and reduces algebra errors.
对于定积分，你必须把上下限从 x 值换成 u 值。另一种方法是在 u 中积分，再换回 x，然后使用原来的上下限。换限通常更快，也能减少代数错误。
Typical substitutions include u = x² + 1, u = ln x, u = sin x, and u = eˣ for a function and its derivative appearing together, possibly up to a constant multiple.
常见换元包括 u = x² + 1、u = ln x、u = sin x 和 u = eˣ。寻找一个函数及其导数同时出现的情况，可能相差一个常数倍。
5. Integration by Parts | 分部积分法
Integration by parts reverses the product rule. It is used for products such as x eˣ, x sin x, ln x, and x ln x. The formula is written in terms of u and dv/dx.
分部积分法是乘积法则的逆运算。它用于 x eˣ、x sin x、ln x 和 x ln x 等乘积形式。公式用 u 和 dv/dx 表示。
∫ u (dv/dx) dx = uv - ∫ v (du/dx) dx
Choose u using the priority order LATE: Logarithmic, Algebraic, Trigonometric, Exponential. The function earlier in this list should usually be u; the remaining part becomes dv/dx.
用优先顺序 LATE 选择 u：对数函数、代数函数、三角函数、指数函数。列表中越靠前的函数通常选作 u，剩余部分作为 dv/dx。
Some integrals require integration by parts twice, for example ∫ eˣ sin x dx or ∫ x² eˣ dx. If the original integral reappears, rearrange the equation to solve for it.
有些积分需要分部积分两次，例如 ∫ eˣ sin x dx 或 ∫ x² eˣ dx。如果原积分重新出现，就重新排列方程来解出它。
= ln x and dv/dx = 1, giving x ln x - x + C. This is a standard result worth remembering.
对于 ∫ ln x dx，令 u = ln x，dv/dx = 1，得到 x ln x - x + C。这是一个值得记住的标准结果。
6. Partial Fractions and Rational Functions | 部分分式与有理函数积分
Rational functions often become integrable after partial fraction decomposition. Factorise the denominator first, then split the fraction into simpler terms with unknown constants.
有理函数在部分分式分解后通常变得可积。先分解分母，再把分式拆成带有未知常数的简单项。
For a linear factor (ax + b), use a term A/(ax + b). For a repeated factor (ax + b)², use A/(ax + b) + B/(ax + b)². For an irreducible quadratic factor, use (Ax + B)/(quadratic).
对于线性因子 (ax + b)，使用项 A/(ax + b)。对于重复因子 (ax + b)²，使用 A/(ax + b) + B/(ax + b)²。对于不可约二次因子，使用 (Ax + B)/二次多项式。
After finding constants, integrate each term. Terms of the form A/(ax + b) integrate to (A/a) ln |ax + b| + C. Terms with an irreducible quadratic may require completing the square and using arctan.
找到常数后，逐项积分。形如 A/(ax + b) 的项积分为 (A/a) ln |ax + b| + C。含有不可约二次因子的项可能需要配方并使用 arctan。
∫ A/(ax + b) dx = (A/a) ln |ax + b| + C
∫ 1/(x² + a²) dx = (1/a) arctan(x/a) + C
7. Parametric Integration | 参数方程积分
can still find the area under the curve. Use the chain rule to replace dx with (dx/dt) dt.
当曲线由参数方程 x = f(t) 和 y = g(t) 给出时，你仍然可以求曲线下面积。使用链式法则把 dx 替换为 (dx/dt) dt。
Area = ∫ y dx = ∫ y (dx/dt) dt
d the matching t-limits. If the curve crosses the x-axis, split the integral at the relevant t-values.
上下限必须是与 x 上下限对应的 t 值。如果题目给出 x 的上下限，解 x = f(t) 找到匹配的 t 上下限。如果曲线穿过 x 轴，在相应的 t 值处拆分积分。
comes V = π ∫ y² (dx/dt) dt. Square y before multiplying by dx/dt, and again use t-limits.
对于参数曲线的旋转体体积，公式变为 V = π ∫ y² (dx/dt) dt。先平方 y，再乘以 dx/dt，同样使用 t 上下限。
V = π ∫ y² (dx/dt) dt
8. Separable Differential Equations | 可分离变量微分方程
arate the variables so that all y terms are on one side and all x terms are on the other, then integrate both sides.
可分离变量微分方程可以写成 dy/dx = f(x) g(y)。分离变量，使所有 y 项在一边、所有 x 项在另一边，然后两边积分。
∫ 1/g(y) dy = ∫ f(x) dx + C
Remember to include the constant C on one side only. If an initial condition is given, substitute the x and y values to find C. If the question asks for y in terms of x, rearrange to make y the subject.
记得只在一边加上常数 C。如果给出初始条件，代入 x 和 y 值求 C。如果题目要求用 x 表示 y，就重新排列使 y 成为主元。
Common examples include dy/dx = ky, giving y = A eᵏˣ ng y² = x² + C. Always check whether the solution can be written explicitly or implicitly.
常见例子包括 dy/dx = ky，得到 y = A eᵏˣ，以及 dy/dx = x/y，得到 y² = x² + C。始终检查解可以写成显式还是隐式。
9. Areas and Volumes of Revolution | 面积与旋转体体积
egrating y with respect to x. If the curve is below the axis, the integral is negative, so use absolute value for total area.
曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的面积，通过对 y 关于 x 积分得到。如果曲线在 x 轴下方，积分为负，因此求总面积时要取绝对值。
Area = ∫ₐᵇ y dx
the intersection points. Find intersections first, then determine which curve is above the other in the interval.
两条曲线 y = f(x) 和 y = g(x) 之间的面积是交点之间 ∫ (上曲线 - 下曲线) dx。先求交点，再确定区间内哪条曲线在上方。
n rotated about the y-axis, use V = π ∫ x² dy if the equation is given as x in terms of y.
当区域绕 x 轴旋转时，体积为 V = π ∫ y² dx。当绕 y 轴旋转时，如果方程以 x 关于 y 的形式给出，则使用 V = π ∫ x² dy。
V = π ∫ₐᵇ y² dx
V = π ∫ x² dy
Do not forget the factor π and do not forget to square y or x before integrating. These are two of the most common sources of lost marks in volume questions.
不要忘记因子 π，也不要忘记在积分前平方 y 或 x。这是体积题中最常见的两个失分点。
10. Numerical Integration: Trapezium and Simpson's Rules | 数值积分：梯形法则与辛普森法则
When an integral cannot be evaluated exactly, numerical methods estimate its value. The trapezium rule approximates the area using straight-line segments between ordinates.
当积分无法精确计算时，数值方法可以估计其值。梯形法则用纵坐标之间的直线段近似面积。
Tₙ = h/2 [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ]
rip. The trapezium rule usually overestimates for a convex curve and underestimates for a concave curve, depending on the shape.
其中 h = (b - a)/n 是每个条带的宽度。对于凸曲线，梯形法则通常高估；对于凹曲线，则可能低估，具体取决于形状。
Simpson's rule uses quadratic curves through groups of three ordinates and is generally more accurate. The number of strips n must be even.
辛普森法则通过三组纵坐标使用二次曲线，通常更精确。条带数 n 必须为偶数。
Sₙ = h/3 [y₀ + 4(y₁ + y₃ + ...) + 2(y₂ + y₄ + ...) + yₙ]
For both rules, build a table of x and y values, check the strip width is constant, and substitute carefully. Numerical integration questions often award method marks for correct structure even if the final value is slightly wrong.
对于这两种法则，建立 x 和 y 值表格，检查条带宽度是否恒定，并仔细代入。数值积分题即使最终值略有错误，也常因正确结构而获得方法分。
11. Exam Strategy, Accuracy Marks and Common Errors | 考试策略、精度分与常见错误
In Edexcel exams, integration questions test both method and accuracy. Write down the substitution or integration by parts formula you are using, show the intermediate step, and keep your notation consistent.
在爱德思考试中，积分题同时考查方法和精度。写下你使用的换元或分部积分公式，展示中间步骤，并保持符号一致。
Common errors include missing '+ C' on indefinite integrals, forgetting to change limits after substitution, dropping the factor π in volume questions, and using an odd number of strips with Simpson's rule.
常见错误包括不定积分遗漏 '+ C'、换元后忘记换限、体积题漏掉因子 π，以及辛普森法则使用奇数条带。
When a question says 'show that' or 'prove that', you must present a logical chain of steps. When a question says 'hence' or 'hence find', you should use the previous result rather than starting again from scratch.
当题目说 'show that' 或 'prove that' 时，你必须展示逻辑步骤链。当题目说 'hence' 或 'hence find' 时，你应当使用前一个结果，而不是从头重新开始。
Check whether the answer should be exact or decimal. Edexcel often requires exact values involving ln, π, e, or surds unless the question specifies a decimal place. Do not round too early in your working.
检查答案应当精确还是小数。爱德思通常要求包含 ln、π、e 或根式的精确值，除非题目指定小数位。不要在计算过程中过早四舍五入。
12. Mixed Practice and Formula Checklist | 综合练习与公式清单
Use this checklist to revise efficiently. For each technique, practise at least two exam-style questions and one past-paper question under timed conditions.
用这份清单高效复习。对每种方法，至少练习两道考试风格题和一道限时真题。
- Reverse differentiation and '+ C' | 逆向微分与 '+ C'
- Standard integrals and linearity | 标准积分与线性性质
- Definite integrals and signed area | 定积分与有向面积
- Substitution, including changing limits | 换元积分，包括换限
- Integration by parts and LATE | 分部积分与 LATE
- Partial fractions for rational functions | 有理函数的部分分式
- Parametric area and volume | 参数方程面积与体积
- Separable differential equations | 可分离变量微分方程
- Area between curves and volumes of revolution | 曲线间面积与旋转体体积
- Trapezium rule and Simpson's rule | 梯形法则与辛普森法则
Memorise the standard results, but more importantly, recognise which technique a question is asking for. The form of the integrand - product, quotient, composition, or parametric - usually tells you the method.
记住标准结果，但更重要的是识别题目要求哪种方法。被积函数的形式——乘积、
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