Integration: Key Concepts for A-Level Maths | A-Level 数学：积分考点精讲

📚 Integration: Key Concepts for A-Level Maths | A-Level 数学：积分考点精讲

Integration is one of the two central pillars of calculus, the reverse process of differentiation. In A‑Level Mathematics, a confident command of integration techniques is essential not only for pure mathematics proofs but also for tackling applied problems in mechanics, probability, and statistics. This revision guide gathers the key concepts, standard results, and exam‑ready strategies you need to master indefinite and definite integrals, substitution, integration by parts, partial fractions, and applications such as areas under curves and volumes of revolution.

积分是微积分的两大核心支柱之一，是微分的逆运算。在 A‑Level 数学中，熟练掌握积分技巧不仅是纯数证明的基础，也是解决力学、概率和统计等应用题的关键。本复习指南汇集了不定积分与定积分的核心概念、标准结果和应试策略，涵盖代换法、分部积分、部分分式法以及曲线下面积、旋转体体积等重要考点。

1. Indefinite Integrals and Standard Results | 不定积分与基本积分公式

An indefinite integral of a function f(x) is any function F(x) such that F′(x) = f(x). Because the derivative of a constant is zero, the most general antiderivative is written as ∫ f(x) dx = F(x) + C, where C is an arbitrary constant of integration.

不定积分就是求导数的逆过程。如果 F′(x) = f(x)，则 ∫ f(x) dx = F(x) + C，其中 C 为任意常数。积分常数是考试中极易遗漏的细节，在不定积分中必须写出。

You must be fluent with the fundamental integration formula for powers: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. For the special case n = −1, the result is ∫ 1/x dx = ln|x| + C. The table below lists the most common integrals required at A‑Level.

必须熟练掌握幂函数的积分公式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1），而 n = −1 时，∫ 1/x dx = ln|x| + C。下表列出了 A‑Level 最常见的基本积分公式。

f(x)	∫ f(x) dx
xⁿ (n ≠ −1)	xⁿ⁺¹/(n+1) + C
1/x	ln\|x\| + C
eˣ	eˣ + C
sin x	−cos x + C
cos x	sin x + C
sec² x	tan x + C
cosec² x	−cot x + C
sec x tan x	sec x + C
cosec x cot x	−cosec x + C
1/(1+x²)	arctan x + C
1/√(1−x²)	arcsin x + C

Linear adjustments inside the integrand are handled by noting that ∫ f(ax + b) dx = (1/a) F(ax + b) + C, provided F is an antiderivative of f. For example, ∫ cos(2x + 1) dx = ½ sin(2x + 1) + C.

当被积函数是线性函数时，可利用 ∫ f(ax + b) dx = (1/a) F(ax + b) + C 快速求解，其中 F 是 f 的一个原函数。例如 ∫ cos(2x + 1) dx = ½ sin(2x + 1) + C。

2. Integration by Substitution | 代换积分法

Substitution is the most versatile technique for simplifying an integral. Given an integral of the form ∫ g(f(x)) f′(x) dx, we set u = f(x) so that du = f′(x) dx, transforming the integral into ∫ g(u) du. For definite integrals, the limits must also be changed to u‑values.

代换法是最灵活的积分技巧。对于形如 ∫ g(f(x)) f′(x) dx 的积分，令 u = f(x)，则 du = f′(x) dx，原积分即变为 ∫ g(u) du。对于定积分，上下限也必须替换为 u 的对应值。

When the substitution is provided, simply differentiate the given u, express dx in terms of du, rewrite the entire integrand in u, and integrate. If the substitution is not given, look for an “inner” function whose derivative appears elsewhere in the integrand. A classic example is ∫ 2x √(x²+1) dx: let u = x²+1, then du = 2x dx, giving ∫ √u du = ⅔ u³⁄² + C = ⅔ (x²+1)³⁄² + C.

若题目给出了代换式，只需对 u 求导，用 du 表达 dx，将整个被积函数写成 u 的函数再积分。若未给代换，则需要寻找一个“内层函数”，其导数恰好在被积函数中出现。典型例子：∫ 2x √(x²+1) dx，令 u = x²+1，则 du = 2x dx，得到 ∫ √u du = ⅔ u³⁄² + C = ⅔ (x²+1)³⁄² + C。

For definite integrals, after substituting and changing limits, there is no need to return to the original variable. Always write the new limits clearly and evaluate directly.

对于定积分，代换后同步变换上下限，无需再换回原变量。务必清晰地写出新的积分限，直接代入计算。

3. Integration by Parts | 分部积分法

Integration by parts comes from the product rule for differentiation. The formula is ∫ u dv = uv − ∫ v du, where u and dv are chosen from the original integrand. The choice of u and dv should follow the LIATE rule of thumb: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential – giving priority to u in that order.

分部积分法源于乘法求导法则，公式为 ∫ u dv = uv − ∫ v du。选择 u 和 dv 时，通常遵循 LIATE 优先级顺序：对数函数、反三角函数、代数函数、三角函数、指数函数，优先将排在前面的函数选为 u。

A typical case is ∫ x eˣ dx. Let u = x, dv = eˣ dx, so du = dx, v = eˣ. Then ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C. If the product involves ln x, always set u = ln x, because its derivative 1/x is simpler.

典型例子 ∫ x eˣ dx，设 u = x，dv = eˣ dx，则 du = dx，v = eˣ，得到 ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。若被积函数含有 ln x，应优先令 u = ln x，因为其导数 1/x 更简单。

Sometimes two applications of integration by parts are needed, such as ∫ x² sin x dx, or the technique may lead back to the original integral, setting up an equation to solve for the unknown integral – common with ∫ eˣ sin x dx.

有时需要连续两次分部积分，如 ∫ x² sin x dx；也可能在运算中重现原积分，从而通过解方程求得结果，这常见于 ∫ eˣ sin x dx 这类题目。

4. Trigonometric Integrals | 三角函数的积分

Integrals of trigonometric functions often require the use of identities to reduce powers or to break products into sums. Essential identities include sin²x = ½(1 − cos 2x), cos²x = ½(1 + cos 2x), and sin 2x = 2 sin x cos x.

三角函数的积分通常需要借助恒等式降幂或将乘积化为和差。核心恒等式包括 sin²x = ½(1 − cos 2x)，cos²x = ½(1 + cos 2x)，以及 sin 2x = 2 sin x cos x。

For integrals of the form ∫ sinᵐ x cosⁿ x dx, if either m or n is odd, reserve one factor and convert the rest using sin²x + cos²x = 1, then apply a simple substitution like u = cos x or u = sin x. If both powers are even, use the double‑angle identities to reduce the exponents.

对于 ∫ sinᵐ x cosⁿ x dx 型积分，若 m 或 n 为奇数，可保留一个因子，其余用 sin²x + cos²x = 1 转换，然后通过 u = cos x 或 u = sin x 代入。若 m 和 n 均为偶数，则需用二倍角公式反复降幂。

Products of sines and cosines with different arguments, like ∫ sin(3x) cos(2x) dx, are integrated by expressing the product as a sum: sin A cos B = ½[sin(A+B) + sin(A−B)]. Similar transform formulas exist for sin A sin B and cos A cos B.

对于不同系数的正弦余弦乘积，例如 ∫ sin(3x) cos(2x) dx，可利用积化和差公式：sin A cos B = ½[sin(A+B) + sin(A−B)]。sin A sin B 和 cos A cos B 也有相应的和差公式。

5. Integration Using Partial Fractions | 部分分式积分法

Rational functions where the numerator has lower degree than the denominator can be split into simpler partial fractions. The resulting integrals typically involve natural logarithms or inverse tangent functions. Always start by factorising the denominator completely.

当有理函数的分子次数低于分母次数时，可将其拆分为更简单的部分分式，所得积分通常涉及自然对数或反正切函数。首先必须将分母彻底因式分解。

For a linear factor (ax + b), write a term A/(ax + b). For a repeated linear factor (ax + b)², include A/(ax + b) + B/(ax + b)². For an irreducible quadratic factor (ax² + bx + c), write a numerator of the form (Px + Q)/(ax² + bx + c), which may lead to a natural logarithm after completing the square or to an arctan term.

对于一次因子 (ax + b)，设部分分式 A/(ax + b)；对于重因子 (ax + b)²，需设 A/(ax + b) + B/(ax + b)²；对于不可约二次因子 (ax² + bx + c)，分子设为 (Px + Q)/(ax² + bx + c)，配方后可转化为对数或 arctan 形式。

Once the constants A, B, P, Q are determined, each term integrates quickly: ∫ A/(ax + b) dx = (A/a) ln|ax + b| + C, and ∫ (Px + Q)/(x² + a²) – after suitable manipulation – gives a logarithmic part and an arctan part.

确定待定常数后，各项可直接积分：∫ A/(ax + b) dx = (A/a) ln|ax + b| + C；而对于 ∫ (Px + Q)/(x² + a²) 经拆分可得到对数和 arctan 的组合。

6. Definite Integrals and the Fundamental Theorem of Calculus | 定积分与微积分基本定理

The definite integral ∫ₐᵇ f(x) dx calculates the net signed area between the curve y = f(x) and the x‑axis from x = a to x = b. The Fundamental Theorem links differentiation and integration: if F′(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a).

定积分 ∫ₐᵇ f(x) dx 计算曲线 y = f(x) 与 x 轴在区间 [a, b] 上的带符号净面积。微积分基本定理揭示了微分与积分的联系：若 F′(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。

Be careful with areas when the curve crosses the x‑axis: integrate separately for positive and negative sections, taking absolute values if the total area is required. The property ∫ₐᵇ f(x) dx = −∫ᵦₐ f(x) dx is useful when limits are reversed.

当曲线穿过 x 轴时，需分段积分，正区域和负区域应分别处理；若求总面积，则对每部分取绝对值。积分限反转的性质 ∫ₐᵇ f(x) dx = −∫ᵦₐ f(x) dx 也常用来调整积分方向。

Key properties include linearity: ∫ₐᵇ [k f(x) ± g(x)] dx = k∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx; and additivity: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx for any c between a and b.

定积分的重要性质包括线性性质：∫ₐᵇ [k f(x) ± g(x)] dx = k∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx；以及区间可加性：若 c 介于 a 与 b 之间，则 ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx。

7. Areas Under Curves | 曲线下面积的计算

To find the area bounded by a curve y = f(x), the x‑axis, and the vertical lines x = a and x = b, compute A = ∫ₐᵇ |f(x)| dx if f(x) changes sign. In practice, evaluate the integral of the top function minus the bottom function when finding the area between two curves: A = ∫ₐᵇ (yᵤppₑᵣ − yₗₒwₑᵣ) dx.

计算由曲线 y = f(x)、x 轴及直线 x = a、x = b 所围区域的面积时，若 f(x) 变号，需用 A = ∫ₐᵇ |f(x)| dx。当求两条曲线之间的面积时，使用“上减下”公式：A = ∫ₐᵇ (y_上 − y_下) dx。

Always sketch the region or check the intersection points, because missing negative parts or mixing the order of subtraction are common mistakes. If the area is bounded by parametric curves, use A = ∫ y (dx/dt) dt with appropriate limits.

建议先画出区域的示意图或求出交点，因为遗漏负值区域或混淆上下曲线顺序是常见错误。如果曲线由参数方程给出，可用 A = ∫ y (dx/dt) dt，并注意积分限的对应关系。

8. Volumes of Revolution | 旋转体体积

When a curve y = f(x) is rotated 360° about the x‑axis, the resulting solid’s volume is V = π ∫ₐᵇ y² dx. If rotation is about the y‑axis, the formula becomes V = π ∫ₐᵇ x² dy, where the function must first be rearranged to give x in terms of y.

当曲线 y = f(x) 绕 x 轴旋转一周时，所得旋转体的体积为 V = π ∫ₐᵇ y² dx。若绕 y 轴旋转，则使用 V = π ∫ₐᵇ x² dy，此时需先将方程改写为 x 关于 y 的函数。

For a region between two curves rotated about an axis, the volume of revolution uses the difference of squares: V = π ∫ₐᵇ (y_outer² − y_inner²) dx. Remember that the solid must be formed by a full 360° rotation, and the limits are along the axis of rotation.

若区域由两条曲线围成并绕轴旋转，体积公式使用“外壳平方减内壳平方”：V = π ∫ₐᵇ (y_外² − y_内²) dx。务必确认旋转为完整的一周，且积分限沿旋转轴方向设置。

9. Integration with Parametric Equations | 参数方程下的积分

Given a curve defined by x = f(t), y = g(t), the area under the curve can be found by changing the variable: ∫ y dx = ∫ g(t) f′(t) dt. The limits of integration for t must correspond to the desired x‑values.

对于由参数方程 x = f(t), y = g(t) 定义的曲线，可通过换元法计算曲线下面积：∫ y dx = ∫ g(t) f′(t) dt，其中 t 的积分限需与 x 的区间对应。

Volumes of revolution can also be handled parametrically. For rotation about the x‑axis, V = π ∫ y² (dx/dt) dt. Always ensure that dx/dt does not change sign over the interval to avoid self‑intersection effects.

旋转体体积也可借助参数方程计算。绕 x 轴旋转时，V = π ∫ y² (dx/dt) dt。应注意 dx/dt 在整个区间内不变号，否则会导致自交区域。

10. Integration in Differential Equations | 积分在微分方程中的应用

Separable first‑order differential equations are solved by separating variables and integrating both sides: ∫ (1/g(y)) dy = ∫ h(x) dx. After integration, include a single constant C, then use any initial conditions to find its value.

可分离变量的一阶微分方程通过分离变量并对两边积分求解：∫ (1/g(y)) dy = ∫ h(x) dx。积分后只加一个常数 C，再代入初始条件求出具体数值。

The A‑Level syllabus also covers integrating factor methods for linear ODEs, but the essential integration skill remains: after multiplying by the integrating factor, you recognise the left‑hand side as the derivative of a product and then integrate directly.

A‑Level 大纲还包括线性常微分方程的积分因子法，但其核心积分技巧不变：乘上积分因子后，左边被识别为某个乘积的导数，然后直接积分。

11. Improper Integrals | 反常积分简介

An improper integral occurs when the interval of integration is infinite, or when the integrand has a vertical asymptote within the interval. Such integrals are evaluated by limiting processes: ∫ₐ∞ f(x) dx = lim_{b→∞} ∫ₐᵇ f(x) dx. If the limit exists and is finite, the integral converges; otherwise, it diverges.

当积分区间为无穷大，或被积函数在区间内有垂直渐近线时，即为反常积分。这类积分通过极限过程求值：∫ₐ∞ f(x) dx = lim_{b→∞} ∫ₐᵇ f(x) dx。若极限存在且有限，则积分收敛；否则发散。

Common examples include ∫₁∞ 1/x² dx (converges) and ∫₁∞ 1/x dx (diverges). Recognising the behaviour near singularities or towards infinity is essential for volume and area problems that stretch to infinity.

常见的例子如 ∫₁∞ 1/x² dx（收敛）和 ∫₁∞ 1/x dx（发散）。在面积或体积问题中，区间延伸到无穷时，必须能判断反常积分的收敛性。

12. Common Exam Pitfalls and Strategic Tips | 常见错误与应试技巧

Always add the constant of integration +C for indefinite integrals, even if the question focuses on technique. For definite integrals, remember to apply the new limits after substitution and never leave the constant in the final answer.

不定积分中务必加上常数 C，即使题目侧重点在于技巧。定积分中，代换后要同步更换积分限，最终答案中绝不要保留常数项。

When using integration by parts, a poor choice of u and dv can make the integral more complicated. If you find yourself stuck, restart with a different choice. For partial fractions, double‑check your decomposition by combining the fractions back to a common denominator.

使用分部积分时，若 u 和 dv 选取不当，积分会变得更复杂。如果陷入困境，重新选 u 再试。用部分分式时，可通过通分还原检验拆分是否正确。

Practice tracing the sign of areas when curves cross the axis, and always sketch the region for area/volume problems. Time management in exams is crucial: learn to recognise which method a given integrand suggests, and be prepared to combine techniques – for example, substitution followed by partial fractions.

当曲线穿越坐标轴时，要练习判断面积的正负号，面对面积或体积问题时务必画草图。考试中时间管理至关重要：应能快速识别被积函数所暗示的解法，并做好综合运用多种技巧的准备，例如先用代换、再用部分分式。

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Integration: Key Concepts for A-Level Maths | A-Level 数学：积分 考点精讲