A-Level CCEA Mathematics: Integration Key Points | A-Level CCEA 数学：积分考点精讲

📚 A-Level CCEA Mathematics: Integration Key Points | A-Level CCEA 数学：积分考点精讲

Integration is a cornerstone of A-Level CCEA Mathematics, bridging differentiation and enabling the calculation of areas, volumes, and solutions to differential equations. Mastering integration requires fluency in standard integrals, techniques like substitution and parts, and the ability to apply integrals to real-world problems. This revision guide consolidates the essential integration concepts tested in CCEA papers, with clear formulas, worked examples, and exam tips.

积分是 A-Level CCEA 数学的基石，它连接了微分并能够计算面积、体积以及微分方程的解。掌握积分需要熟练掌握标准积分公式、换元法和分部积分等技巧，并能将积分应用于实际问题。本复习指南整合了 CCEA 考试中考查的积分核心概念，配有清晰的公式、示例和考试技巧。

1. Indefinite Integration and Basic Rules | 不定积分与基本规则

Indefinite integration is the reverse process of differentiation. If F'(x) = f(x), then ∫ f(x) dx = F(x) + C, where C is the constant of integration.

不定积分是微分的逆过程。如果 F'(x) = f(x)，那么 ∫ f(x) dx = F(x) + C，其中 C 是积分常数。

The power rule for integration: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, provided n ≠ −1. For n = −1, we have ∫ x⁻¹ dx = ln|x| + C.

幂函数的积分法则：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，要求 n ≠ −1。当 n = −1 时，∫ x⁻¹ dx = ln|x| + C。

The integral of a sum or difference is the sum or difference of the integrals: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.

函数和或差的积分等于各函数积分的和或差：∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。

Always remember to include the constant of integration ‘+ C’ when evaluating an indefinite integral. Missing it is a common mistake that loses marks in CCEA exams.

在计算不定积分时，务必记得加上积分常数 ‘+ C’。忘记这一点是 CCEA 考试中常见的失分错误。

2. Table of Basic Integrals | 基本积分公式表

Memorising the following standard integrals is essential for speed and accuracy in the exam. They are the building blocks for more complex problems.

熟记下列标准积分公式对考试中的速度和准确性至关重要。它们是解决更复杂问题的基石。

f(x)	∫ f(x) dx
xⁿ (n ≠ −1)	(xⁿ⁺¹)/(n+1) + C
1/x	ln\|x\| + C
eˣ	eˣ + C
aˣ (a > 0, a ≠ 1)	aˣ / ln a + C
sin x	−cos x + C
cos x	sin x + C
sec² x	tan x + C
cosec x cot x	−cosec x + C
sec x tan x	sec x + C
1/√(a² − x²)	sin⁻¹(x/a) + C, \|x\| < a
1/(a² + x²)	(1/a) tan⁻¹(x/a) + C

For linear substitutions inside a function, you can use the reverse chain rule: ∫ f ‘(ax + b) dx = (1/a) f(ax + b) + C. This is particularly useful for integrals like ∫ e²ˣ dx = (1/2)e²ˣ + C and ∫ cos(3x) dx = (1/3)sin(3x) + C.

对于函数内部的线性代换，可以使用反链式法则：∫ f ‘(ax + b) dx = (1/a) f(ax + b) + C。这对诸如 ∫ e²ˣ dx = (1/2)e²ˣ + C 和 ∫ cos(3x) dx = (1/3)sin(3x) + C 的积分特别有用。

3. Definite Integrals and Area | 定积分与面积

A definite integral ∫ₐᵇ f(x) dx evaluates the exact signed area between the curve y = f(x) and the x-axis from x = a to x = b. If F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a).

定积分 ∫ₐᵇ f(x) dx 计算的是曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的精确有向面积。如果 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。

Areas above the x-axis give a positive contribution; areas below the x-axis give a negative contribution. If the curve crosses the x-axis, you must split the integral at the roots and sum the absolute values to find the total area.

x 轴上方的面积为正，x 轴下方的面积为负。如果曲线与 x 轴相交，则必须在交点处拆分积分，并取各段绝对值之和来求得总面积。

Always remember to write the limits clearly and substitute correctly. A common error is forgetting to change signs when subtracting the lower limit.

务必清晰地写出上下限并正确代入。一个常见的错误是在减去下限时忘记变号。

4. Area between Curve and Axes | 曲线与坐标轴围成的面积

To find the area enclosed by a curve and the x-axis between two limits, first identify where f(x) = 0 within the interval. Then calculate Total Area = ∫ₐᶜ |f(x)| dx by integrating separately in regions where f(x) is positive and where it is negative, taking the absolute value of each integral.

要计算曲线与 x 轴在给定区间内所围的面积，首先要找出区间内 f(x)=0 的点。然后通过分别在 f(x) 为正和为负的区域积分，并将每个积分的绝对值相加来计算总面积 = ∫ₐᶜ |f(x)| dx。

If the curve is given in the form x = g(y), the area between the curve and the y-axis from y = c to y = d is given by ∫ₚᵠ |g(y)| dy. The same principle of splitting at y-intercepts applies.

如果曲线以 x = g(y) 的形式给出，曲线与 y 轴之间从 y = c 到 y = d 的面积为 ∫ₚᵠ |g(y)| dy。同样的，在 y 截距处分割积分的原则也适用。

5. Area between Two Curves | 两曲线间的面积

The area between two curves y = f(x) and y = g(x) from x = a to x = b, where f(x) ≥ g(x) on [a, b], is A = ∫ₐᵇ [f(x) − g(x)] dx.

两条曲线 y = f(x) 和 y = g(x) 在区间 [a, b] 上且 f(x) ≥ g(x) 时，它们之间的面积是 A = ∫ₐᵇ [f(x) − g(x)] dx。

Always start by finding the points of intersection of the two curves, as these will serve as the limits of integration. If the curves cross, you must set up separate integrals for each region where the ‘upper’ and ‘lower’ curves swap.

始终先从求出两曲线的交点开始，这些交点将作为积分的上下限。如果曲线相交，你必须为“上”“下”曲线互换的每个区域分别设置积分。

The formula ‘top minus bottom’ integrates with respect to x, while ‘right minus left’ integrates with respect to y. Choose the orientation that makes the integration easier.

“上减下”公式是对 x 积分，而“右减左”是对 y 积分。选择能让积分更简单的方向。

6. Integration by Substitution | 换元法

Integration by substitution is used to simplify an integral by replacing a complicated inner function with a single variable. If we let u = g(x), then du/dx = g'(x), giving dx = du / g'(x). The integral ∫ f(g(x)) g'(x) dx becomes ∫ f(u) du.

换元法通过用一个新变量替换复杂的内层函数来简化积分。如果设 u = g(x)，则 du/dx = g'(x)，于是 dx = du / g'(x)。积分 ∫ f(g(x)) g'(x) dx 就变成了 ∫ f(u) du。

For definite integrals, you must also change the limits of integration. If x goes from a to b, then u goes from g(a) to g(b). This avoids having to convert back to x at the end.

对于定积分，必须同时改变积分的上下限。如果 x 从 a 变到 b，那么 u 就从 g(a) 变到 g(b)。这样在最后就不用再换回 x 了。

Example: Evaluate ∫ 2x √(x² + 1) dx. Let u = x² + 1, then du/dx = 2x, so dx = du/(2x). Substituting gives ∫ √u du = (2/3)u³/² + C = (2/3)(x² + 1)³/² + C.

示例：计算 ∫ 2x √(x² + 1) dx。设 u = x² + 1，则 du/dx = 2x，所以 dx = du/(2x)。代入后得到 ∫ √u du = (2/3)u³/² + C = (2/3)(x² + 1)³/² + C。

7. Integration by Parts | 分部积分法

Integration by parts is derived from the product rule for differentiation. The formula is: ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. It is useful for integrals involving products of functions, such as x eˣ, x sin x, or ln x.

分部积分法源于微分的乘法法则。公式为：∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx。它对涉及函数乘积的积分非常有用，例如 x eˣ, x sin x 或 ln x。

When choosing u and dv, remember the LIATE rule of thumb: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Functions appearing earlier in this list are usually good choices for u.

在选择 u 和 dv 时，记住 LIATE 经验法则：对数函数、反三角函数、代数函数、三角函数、指数函数。列表中越靠前的函数通常越适合选作 u。

Example: Find ∫ x sin x dx. Let u = x (Algebraic) and dv/dx = sin x. Then du/dx = 1 and v = −cos x. Applying the formula gives −x cos x + ∫ cos x dx = −x cos x + sin x + C.

示例：求 ∫ x sin x dx。设 u = x（代数函数）且 dv/dx = sin x。则 du/dx = 1，v = −cos x。应用公式得 −x cos x + ∫ cos x dx = −x cos x + sin x + C。

For definite integrals, apply the limits to the [uv] term and the remaining integral: ∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du.

对于定积分，将上下限应用于 [uv] 项和剩余的积分：∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du。

8. Using Initial Conditions to Find the Constant of Integration | 使用初始条件求积分常数

When a gradient function dy/dx is given together with a point (x₀, y₀) that lies on the curve, you can find the equation of the curve by integrating and then using the point to solve for C.

当给出导数函数 dy/dx 以及曲线上一点 (x₀, y₀) 时，你可以通过积分，然后利用该点求解 C 来找到曲线方程。

Integrate dy/dx to obtain y = F(x) + C. Substitute the coordinates x = x₀, y = y₀ into the equation and rearrange to find C. Finally, write the full equation y = F(x) + C.

对 dy/dx 积分得到 y = F(x) + C。将坐标 x = x₀，y = y₀ 代入方程，整理求出 C。最后写出完整的方程 y = F(x) + C。

This process is very common in kinematics where velocity or acceleration is integrated to find displacement, given initial conditions.

这个过程在运动学中非常常见，即给定初始条件，对速度或加速度积分以求位移。

9. Integrating Parametric Equations | 参数方程积分

For a curve defined parametrically as x = f(t), y = g(t), the area under the curve between t = t₁ and t = t₂ is given by A = ∫ y dx = ∫ g(t) f ‘(t) dt, where the limits are the t-values corresponding to the required x-interval.

对于由参数方程 x = f(t), y = g(t) 定义的曲线，曲线下方在 t = t₁ 到 t = t₂ 之间的面积为 A = ∫ y dx = ∫ g(t) f ‘(t) dt，其中积分限是相应 x 区间对应的 t 值。

It is essential to change the limits to t-values. Never mix x-limits with a t-integrand. Also be careful with the direction: if t increases but x decreases, the integral will naturally give a signed area; adjust signs if total area is required.

将积分限换成 t 值至关重要。切勿将 x 的积分限与关于 t 的被积函数混用。还要注意方向：如果 t 增加而 x 减小，积分自然会给出有向面积；如果需要总面积则要调整符号。

10. Volume of Revolution | 旋转体体积

When a region under a curve y = f(x) between x = a and x = b is rotated 360° about the x-axis, a solid of revolution is formed. Its volume is V = π ∫ₐᵇ y² dx = π ∫ₐᵇ [f(x)]² dx.

将曲线 y = f(x) 下方从 x = a 到 x = b 的区域绕 x 轴旋转 360°，形成一个旋转体。其体积为 V = π ∫ₐᵇ y² dx = π ∫ₐᵇ [f(x)]² dx。

For rotation about the y-axis, the formula becomes V = π ∫ₚᵠ x² dy, where the limits are y-values. You may need to rearrange the curve’s equation to express x² in terms of y.

对于绕 y 轴旋转，公式变为 V = π ∫ₚᵠ x² dy，其中积分限是 y 值。你可能需要重新整理曲线方程，用 y 表示 x²。

If the region is between two

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A-Level CCEA Mathematics: Integration Key Points | A-Level CCEA 数学：积分 考点精讲