A-Level WJEC Mathematics: Integration Revision Guide | A-Level WJEC 数学：积分 考点精讲

📚 A-Level WJEC Mathematics: Integration Revision Guide | A-Level WJEC 数学：积分 考点精讲

Integration is a core topic in A-Level WJEC Mathematics, extending the process of differentiation to find antiderivatives, areas, volumes, and solutions to differential equations. Mastery of standard integrals, substitution, parts, and applications is essential for success in both Pure and Applied units. This guide systematically covers the key integration concepts, providing bilingual explanations and worked-style statements to reinforce understanding and exam technique.

积分是 A-Level WJEC 数学的核心主题，将微分过程延伸至寻找原函数、面积、体积以及微分方程的解。掌握标准积分、换元法、分部积分及其应用对于纯数与应数单元的成功至关重要。本指南系统地覆盖关键积分概念，提供中英双语解释与例题式陈述，以强化理解与应试技巧。

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1. Indefinite Integration & Standard Forms | 不定积分与标准形式

resents the general antiderivative of f(x), always including an arbitrary constant C because differentiation of a constant yields zero. You must memorise the standard integrals for powers, exponentials, logarithms, and trigonometric functions as they form the foundation for all other techniques.

不定积分 ∫ f(x) dx 表示 f(x) 的全体原函数，由于常数的导数为零，务必加上任意常数 C。必须熟记幂函数、指数函数、对数函数和三角函数的积分标准形式，它们是一切其他技巧的基础。

f(x)	∫ f(x) dx
xⁿ (n ≠ -1)	xⁿ⁺¹/(n+1) + C
1/x	ln |x| + C
eˣ	eˣ + C
cos x	sin x + C
sin x	-cos x + C
sec² x	tan x + C
cosec x cot x	-cosec x + C
sec x tan x	sec x + C

b)ⁿ dx = (1/a) * (ax+b)ⁿ⁺¹/(n+1) + C, provided n ≠ -1. For rational exponents, recall that √x = x^½ and treat it accordingly.

当积分形如 (ax+b)ⁿ 的表达式时，可使用逆链式法则：∫ (ax+b)ⁿ dx = (1/a) · (ax+b)ⁿ⁺¹/(n+1) + C，其中 n ≠ -1。对于有理指数，记住 √x = x^½ 并按幂函数处理。

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2. Integration by Substitution | 换元积分法

a function of x with a new variable u. The formal statement is: let u = g(x), then du = g'(x) dx, so ∫ f(g(x)) g' For definite integrals, the limits must be changed: x = a → u = g(a) and x = b → u = g(b).

换元积分法通过将 x 的函数替换为新变量 u 来简化被积函数。形式化表示为：设 u = g(x)，则 du = g'(x) dx，于是 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。对于定积分，需同时替换上下限：x = a → u = g(a) 且 x = b → u = g(b)。

u = x², then du = 2x dx, turning the integral into ∫ sin u du = -cos u + C = -cos(x²) + C.

WJEC 常见题型是识别出内层函数的导数恰好存在。例如计算 ∫ 2x sin(x²) dx，设 u = x²，则 du = 2x dx，积分变为 ∫ sin u du = -cos u + C = -cos(x²) + C。

When the integrand contains √a² - x², √a² + x², or √x² - a², trigonometric or hyperbolic substitutions may be employed, but these are less frequent at A-Level; focus on standard polynomial and trigonometric substitutions.

当被积函数包含 √a² - x²、√a² + x² 或 √x² - a² 时，可使用三角或双曲换元，但在 A-Level 中出现较少；重点掌握标准多项式与三角换元。

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3. Integration by Parts | 分部积分法

. Selecting u and dv appropriately is key; a common mnemonic is LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to choose u.

分部积分法源自微分的乘法法则，对于积分两个函数的乘积至关重要。公式为 ∫ u (dv/dx) dx = uv - ∫ v (du/dx) dx，常简写为 ∫ u dv = uv - ∫ v du。合理选择 u 和 dv 是关键；常用的 LIATE 口诀（对数、反三角、代数、三角、指数）可帮助决定 u。

Example: ∫ x eˣ dx. Let u = x (algebraic), so dv/dx = eˣ, then du/dx = 1, v = eˣ. Applying the formula gives x eˣ - ∫ eˣ dx = x eˣ - eˣ n x, so that the integral becomes (x²/2) ln x - ∫ (x²/2)(1/x) dx = (x²/2) ln x - x²/4 + C.

示例：∫ x eˣ dx。设 u = x（代数函数），则 dv/dx = eˣ，那么 du/dx = 1，v = eˣ。代入公式得 x eˣ - ∫ eˣ dx = x eˣ - eˣ + C。对于 ∫ x ln x dx，选择 u = ln x，积分为 (x²/2) ln x - ∫ (x²/2)(1/x) dx = (x²/2) ln x - x²/4 + C。

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4. Integrating Partial Fractions | 部分分式积分

Integrating rational functions often requires decomposing them into partial fractions. After expressing a quotient of polynomials as a sum of simpler fractions with linear or repeated factors, you integrate term by term. The resulting integrals typically involve natural logarithms and, in some cases, arctangent functions.

积分有理函数常需将其分解为部分分式。把多项式商表达为线性因子或重因子的简单分式之和后，逐项积分。所得积分通常包含自然对数，有时也出现反正切函数。

For distinct linear factors: write 1/[(x+1)(x-2)] as A/(x+1) + B/(x-2). Then integrate to get A ln|x+1| + B ln|x-2| + C. Repeated factors introduce terms like A/(x-p) + B/(x-p)². For irreducible quadratics, you may obtain an arctan term after completing the square.

对于不重复的线性因子：将 1/[(x+1)(x-2)] 写为 A/(x+1) + B/(x-2)，积分得 A ln|x+1| + B ln|x-2| + C。重因子会引入 A/(x-p) + B/(x-p)² 等形式。对于不可约二次式，配方后可能得到 arctan 项。

WJEC exams frequently test partial fractions combined with integration, especially before finding areas or solving differential equations. Always check the degree of the numerator: if it is equal to or higher than the denominator, perform polynomial long division first.

WJEC 考试常将部分分式与积分结合考查，尤其在求面积或解微分方程之前。务必检查分子次数：若分子次数不低于分母，先进行多项式长除法。

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5. Definite Integrals & Limits | 定积分与上下限

A definite integral ∫ₐ^b x-axis from x = a to x = b. It is evaluated using the Fundamental Theorem of Calculus: ∫ₐ^b f(x) dx = F(b) - F(a), where F'(x) = f(x). Always include the limits when showing working.

定积分 ∫ₐ^b f(x) dx 计算的是从 x = a 到 x = b 曲线与 x 轴之间的带号面积。根据微积分基本定理：∫ₐ^b f(x) dx = F(b) - F(a)，其中 F'(x) = f(x)。解题时必须体现上下限的代入过程。

Properties of definite integrals include: ∫ₐ^a f(x) dx = 0; ∫ₐ^b f(x) dx = -∫_bᵃ f(x) dx; and ∫ₐᶜ f(x) dx = ∫ₐ^b f(x) dx + ∫_bᶜ f(x) dx. Be cautious with functions that cross the x-axis; you may need to split the integral and take absolute values when finding total area.

定积分的性质包括：∫ₐ^a f(x) dx = 0；∫ₐ^b f(x) dx = -∫_bᵃ f(x) dx；以及 ∫ₐᶜ f(x) dx = ∫ₐ^b f(x) dx + ∫_bᶜ f(x) dx。注意当函数穿越 x 轴时，求总面积可能需要分段积分并取绝对值。

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6. Area Under a Curve | 曲线下方面积

[a, b], the area bounded by the curve, the x-axis, and the lines x = a and x = b is A = ∫ₐ^b When f(x) ≤ 0, the integral yields a negative value, so the actual geometrical area is -∫ₐ^b f(x) dx or ∫ₐ^b |f(x)| dx.

若 y = f(x) 在区间 [a, b] 上连续且 f(x) ≥ 0，则由曲线、x 轴和直线 x = a、x = b 围成的面积为 A = ∫ₐ^b f(x) dx。当 f(x) ≤ 0 时，积分值为负，几何面积应为 -∫ₐ^b f(x) dx 或 ∫ₐ^b |f(x)| dx。

o find the area under a parametric curve defined by x = x(t), y = y(t). The area is then ∫_t₁^t₂ y (dx/dt) dt. Remember to convert the limits to t-values using the given parameter interval.

在 WJEC 试题中，可能要求计算由参数方程 x = x(t)，y = y(t) 给出的曲线下方面积。此时面积为 ∫_t₁^t₂ y (dx/dt) dt。记得用给定的参数区间将积分上下限转换为 t 值。

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7. Area Between Curves | 曲线间面积

= f(x) and y = g(x) from x = a to x = b, where f(x) ≥ g(x) across the interval, is A = ∫ₐ^b [f(x) - g(x)] dx. If the curves intersect, you must find the points of intersection and set appropriate limits, integrating the 'top minus bottom' function over each subinterval.

两条曲线 y = f(x) 与 y = g(x) 在 x = a 到 x = b 之间所夹的面积，若在整个区间上 f(x) ≥ g(x)，则 A = ∫ₐ^b [f(x) - g(x)] dx。若曲线相交，须求出交点并设定合适上下限，在每一子区间上对“上方减下方”函数积分。

egions bounded by curves expressed in terms of y, the area is often easier to compute as ∫_c^d [xᵣ(y) - xₗ(y)] dy, where xᵣ is the rightmost curve and xₗ the leftmost. Always sketch the region to avoid sign errors.

对于用 y 表示的曲线所围区域，面积有时更方便计算为 ∫_c^d [x右(y) - x左(y)] dy，其中 x右 为右侧曲线，x左 为左侧曲线。务必画出草图以避免正负号错误。

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8. Volumes of Revolution | 旋转体体积

When a region is rotated about the x-axis, the volume generated is V = π ∫ₐ^b y² dx, where y = f(x). For rotation about the y-axis, the formula becomes V = π ∫_c^d x² dy, with x expressed in terms of y. WJEC frequently tests these with both explicit and parametric equations.

当区域绕 x 轴旋转，生成的体积为 V = π ∫ₐ^b y² dx，其中 y = f(x)。绕 y 轴旋转时，公式变为 V = π ∫_c^d x² dy，需将 x 用 y 表达。WJEC 常结合显式方程与参数方程考查这些内容。

nd the volume of revolution for a parametric curve: for rotation about the x-axis, V = π ∫_t₁^t₂ ² (dx/dt) dt. Similarly, for rotation about the y-axis using parametric equations, V = π ∫_t₁^t₂ x² (dy/dt) dt. Accurate limit conversion is critical.

做好计算参数曲线旋转体体积的准备：绕 x 轴旋转时，V = π ∫_t₁^t₂ y² (dx/dt) dt；类似地，使用参数方程绕 y 轴旋转时，V = π ∫_t₁^t₂ x² (dy/dt) dt。准确转换积分限至关重要。

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9. Solving Differential Equations | 解微分方程

sides to obtain an implicit or explicit solution. The arbitrary constant is often determined using initial conditions given in the problem.

一阶可分离变量的微分方程是积分的一个重要应用。给定 dy/dx = f(x) g(y)，分离变量得 (1/g(y)) dy = f(x) dx，然后两边积分即得隐式或显式解。常需利用题目给出的初始条件确定任意常数。

integrate, and use given data to find k. Always check that your answer satisfies the original differential equation.

WJEC 试题常将微分方程置于实际背景中，如人口增长、冷却或混合问题。先将描述转化为 dy/dx = k(...) 的形式，积分后用给定数据求出 k。最后务必验证答案满足原微分方程。

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10. Numerical Integration: Trapezium Rule | 数值积分：梯形法则

timated as A ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ], where yᵢ = f(xᵢ) and xᵢ = a + i h.

当定积分无法解析计算时，梯形法则可提供近似值。将区间等分为 n 个宽度为 h = (b - a)/n 的条带，面积估算为 A ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ]，其中 yᵢ = f(xᵢ)，xᵢ = a + i h。

The WJEC specification often includes a question on the trapezium rule, working with tabulated values or a given function. Be prepared to state the number of ordinates and strips, and understand that the approximation improves as n increases. You may be asked to compare the approximate value with an exact value found through integration.

WJEC 大纲常包含梯形法则的考题，基于表格数据或给定函数。准备好说明纵坐标与条带的数量，并理解随 n 增大近似值更为精确。有时会要求将近似值与通过积分求得的精确值进行比较。

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