📚 AS Mathematics: Integration Key Points | AS 数学：积分 考点精讲
Welcome to this AS Mathematics revision guide on integration. Mastering integration is essential for calculus and its applications. This article covers all key concepts, from indefinite integrals to area calculations and kinematics.
欢迎阅读本篇 AS 数学积分复习指南。掌握积分对微积分及其应用至关重要。本文涵盖从不定积分到面积计算和运动学应用的所有核心考点。
1. What is Integration? | 什么是积分？
Integration is the reverse process of differentiation. Given a derivative f'(x), we seek the original function F(x) such that F'(x) = f(x). This original function is called an antiderivative or indefinite integral and is denoted by ∫ f(x) dx = F(x) + C, where C is the constant of integration.
积分是微分的逆运算。已知导数 f'(x)，我们要寻找原函数 F(x) 使得 F'(x) = f(x)。这个原函数称为不定积分，记作 ∫ f(x) dx = F(x) + C，其中 C 是积分常数。
Since differentiation eliminates constant terms, we add an arbitrary constant C to account for all possible antiderivatives.
由于微分运算会消去常数项，我们需要加上任意常数 C 来表示所有可能的原函数。
2. Basic Integration Formulae | 基本积分公式
The following table summarises the standard indefinite integrals you must know for AS Mathematics. All results should include the constant of integration C.
下表总结了 AS 数学中必须掌握的标准不定积分。所有结果都应包含积分常数 C。
| f(x) | ∫ f(x) dx |
| xⁿ (n ≠ –1) | (1/(n+1)) xⁿ⁺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 |
Remember that the power rule does not apply when n = –1; instead you get the natural logarithm. For trigonometric integrals, pay attention to signs – the integral of sin x is negative cosine.
请记住，当 n = –1 时不能使用幂法则，此时得到的是自然对数。对于三角函数的积分，要特别注意符号——正弦函数的积分是负余弦。
3. Integration of Linear Functions | 线性函数的积分
When the integrand is a function of a linear expression (ax + b), we can use a simple reverse chain rule. For example, ∫ (ax + b)ⁿ dx = (1/a) · (1/(n+1)) (ax + b)ⁿ⁺¹ + C, provided n ≠ –1. Similarly, ∫ eax+b dx = (1/a) eax+b + C, and ∫ sin(ax+b) dx = –(1/a) cos(ax+b) + C.
当被积函数是线性表达式 (ax + b) 的函数时，我们可以使用简单的逆向链式法则。例如，∫ (ax + b)ⁿ dx = (1/a)·(1/(n+1)) (ax + b)ⁿ⁺¹ + C（前提 n ≠ –1）。类似地，∫ eax+b dx = (1/a) eax+b + C，以及 ∫ sin(ax+b) dx = –(1/a) cos(ax+b) + C。
Always divide by the coefficient of x inside the function. This is because if you differentiate the result, the inner derivative a will appear and require a factor of 1/a to cancel.
一定要除以函数内部 x 的系数。这是因为如果你对结果微分，内部导数 a 会出现，需要通过乘以 1/a 来抵消。
4. Properties of Indefinite Integrals | 不定积分的性质
Integration is a linear operation. The integral of a sum is the sum of the integrals, and a constant factor can be taken outside the integral sign: ∫ [k·f(x) ± g(x)] dx = k ∫ f(x) dx ± ∫ g(x) dx.
积分是线性运算。和的积分等于积分之和，常数因子可以提到积分号外面：∫ [k·f(x) ± g(x)] dx = k ∫ f(x) dx ± ∫ g(x) dx。
These properties allow you to integrate term by term, which is especially useful for polynomials and rational expressions split into partial fractions.
这些性质允许你逐项积分，这对于多项式和拆分为部分分式的有理表达式尤为有用。
5. Definite Integrals and the Fundamental Theorem | 定积分与基本定理
The definite integral ∫ab f(x) dx represents the signed area under the curve y = f(x) from x = a to x = b. The Fundamental Theorem of Calculus links differentiation and integration: if F is an antiderivative of f, then ∫ab f(x) dx = F(b) – F(a).
定积分 ∫ab f(x) dx 表示曲线 y = f(x) 从 x = a 到 x = b 下方带符号的面积。微积分基本定理将微分与积分联系起来：如果 F 是 f 的一个原函数，则 ∫ab f(x) dx = F(b) – F(a)。
When evaluating, always write the antiderivative in square brackets with the limits, then substitute. A common notation is [F(x)]ab = F(b) – F(a).
求值时，始终将原函数写在带有上下限的中括号里，然后代入数值。常见记法为 [F(x)]ab = F(b) – F(a)。
6. Area Under a Curve | 曲线下的面积
To find the area between a curve y = f(x), the x‑axis, and the lines x = a and x = b, compute the definite integral ∫ab f(x) dx, provided f(x) ≥ 0 on [a, b]. If f(x) dips below the x‑axis, the integral yields a negative value for that part, so you must split the interval and take absolute values for total area.
要计算曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 围成的面积，计算定积分 ∫ab f(x) dx，前提是在 [a, b] 上 f(x) ≥ 0。如果 f(x) 落到 x 轴下方，积分会给出负值，因此必须分割区间并对面积取绝对值再求和。
For total area, use ∫ab |f(x)| dx or integrate each piece separately and sum the absolute areas.
求总面积时，可使用 ∫ab |f(x)| dx，或分段积分并取各段绝对面积之和。
7. Area Between Curves | 曲线之间的面积
When two curves intersect, the area between them from x = a to x = b is given by ∫ab |f(x) – g(x)| dx, assuming f and g are continuous and f(x) ≥ g(x) on the interval. You must determine which function is on top by evaluating the difference or using the intersection points.
当两条曲线相交时，从 x = a 到 x = b 它们之间的面积由 ∫ab |f(x) – g(x)| dx 给出，假设 f 和 g 连续且在区间上 f(x) ≥ g(x)。你需要通过计算差值或利用交点来确定哪条曲线在上方。
First find the interval where f – g ≥ 0; then integrate (f – g). If the curves cross, split the integral at the crossing point.
首先确定 f – g ≥ 0 的区间，然后对 (f – g) 积分。如果曲线有交叉，则在交叉点处分割积分。
8. Integration in Kinematics | 运动学中的积分
In AS Mathematics, integration is widely used in kinematics. Given acceleration a(t), velocity v(t) is ∫ a(t) dt, and displacement s(t) is ∫ v(t) dt. Each integration introduces a constant of integration that can be determined using initial conditions, such as initial
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