📚 Integral Calculus: Basic Concepts and Core Methods | 积分基本概念与核心方法
Calculus is built on two fundamental operations: differentiation and integration. In IB Mathematics, integration is a central topic that appears in both Analysis and Approaches (AA) and Applications and Interpretation (AI), though with different depths. This article introduces the basic concepts of integration and the core methods needed to solve standard problems.
微积分建立在两个基本运算之上：微分与积分。在IB数学中，积分是核心主题，出现在分析与方法（AA）以及应用与解释（AI）两个课程中，但深度不同。本文介绍积分的基本概念以及解决标准问题所需的核心方法。
1. Indefinite Integrals and the Constant of Integration | 不定积分与积分常数
An indefinite integral of a function f(x) is a family of functions F(x) + C, where F'(x) = f(x). The constant C is called the constant of integration, and it appears because the derivative of any constant is zero.
函数 f(x) 的不定积分是函数族 F(x) + C，其中 F'(x) = f(x)。常数 C 称为积分常数，它之所以出现是因为任何常数的导数都是零。
We write ∫ f(x) dx = F(x) + C. The symbol ∫ is the integral sign, and dx indicates that the variable of integration is x.
我们记作 ∫ f(x) dx = F(x) + C。符号 ∫ 是积分号，dx 表示积分变量是 x。
Unlike definite integrals, indefinite integrals do not have upper or lower limits; they represent antiderivatives rather than numbers.
与定积分不同，不定积分没有上下限；它表示原函数，而不是一个数值。
2. Basic Integration Formulas | 基本积分公式
If n ≠ −1, then ∫ xn dx = xn+1/(n+1) + C. For n = −1, ∫ x−1 dx = ln|x| + C.
若 n ≠ −1，则 ∫ xn dx = xn+1/(n+1) + C。当 n = −1 时，∫ x−1 dx = ln|x| + C。
Other standard forms include: ∫ ex dx = ex + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C, and ∫ 1/(1+x²) dx = arctan x + C.
其他标准形式包括：∫ ex dx = ex + C，∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C，∫ sec² x dx = tan x + C，以及 ∫ 1/(1+x²) dx = arctan x + C。
3. Linearity of Integration | 积分的线性性质
Integration is linear: ∫ [af(x) + bg(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx, where a and b are constants. This allows us to integrate term by term.
积分是线性的：∫ [af(x) + bg(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx，其中 a、b 为常数。这使我们能够逐项积分。
Another property is that ∫ f(x) dx = ∫ f(t) dt; the variable is a dummy variable. Changing the variable name does not change the antiderivative.
另一个性质是 ∫ f(x) dx = ∫ f(t) dt；变量是哑变量。改变变量名不改变原函数。
4. Fundamental Theorem of Calculus | 微积分基本定理
The Fundamental Theorem of Calculus connects differentiation and integration. If f is continuous on [a, b] and F is an antiderivative of f, then ∫ab f(x) dx = F(b) − F(a).
微积分基本定理将微分与积分联系起来。若 f 在 [a, b] 上连续，F 是 f 的一个原函数，则 ∫ab f(x) dx = F(b) − F(a)。
Also, if we define G(x) = ∫ax f(t) dt, then G'(x) = f(x). This is the second part of the theorem, often used in problems involving accumulation functions.
此外，若定义 G(x) = ∫ax f(t) dt，则 G'(x) = f(x)。这是定理的第二部分，常用于涉及累积函数的问题。
5. Definite Integrals and Their Properties | 定积分及其性质
A definite integral ∫ab f(x) dx gives the signed net area between the graph of y = f(x) and the x-axis from x = a to x = b. Areas below the axis are counted as negative.
定积分 ∫ab f(x) dx 给出曲线 y = f(x) 与 x 轴从 x=a 到 x=b 之间的有符号净面积。轴下方的面积计为负。
Key properties include: ∫ab f(x) dx = −∫ba f(x) dx, ∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx for any c in [a,b], and if f(x) ≤ g(x) on [a,b], then ∫ab f(x) dx ≤ ∫ab g(x) dx.
主要性质包括：∫ab f(x) dx = −∫ba f(x) dx（上下限交换变号）；对 [a,b] 上任意 c，∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx；若在 [a,b] 上 f(x) ≤ g(x)，则 ∫ab f(x) dx ≤ ∫ab g(x) dx。
6. Integration by Substitution | 换元积分法
Substitution is the reverse of the chain rule. If we set u = g(x), then du = g'(x)dx, and ∫ f(g(x)) g'(x) dx = ∫ f(u) du. After integrating, substitute back u = g(x).
换元法是链式法则的逆运算。设 u = g(x)，则 du = g'(x)dx，于是 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。积分后再代回 u = g(x)。
For example, to find ∫ 2x ex² dx, let u = x², so du = 2x dx. The integral becomes ∫ eu du = eu + C = ex² + C.
例如，求 ∫ 2x ex² dx，令 u = x²，则 du = 2x dx。积分变为 ∫ eu du = eu + C = ex² + C。
For definite integrals, either convert the limits to u-values before integrating, or integrate in
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