📚 Integration Part 1: Indefinite Integrals and Basic Rules | 积分第一部分：不定积分与基本法则
In this first part of integration, we introduce indefinite integrals as the reverse process of differentiation. You will learn the power rule, basic standard integrals, linearity, and how to find a particular solution from an initial condition. These skills are essential for later work on areas under curves and differential equations.
在积分的第一部分，我们将不定积分作为微分的逆运算引入。你将学习幂法则、基本标准积分、线性性质，以及如何利用初始条件求特解。这些技能对于后续学习曲线下面积和微分方程至关重要。
1. What Is Integration? | 什么是积分？
Integration is the process of finding a function when its derivative is given. If F'(x) = f(x), we say that F is an antiderivative of f. The set of all antiderivatives is called the indefinite integral of f.
积分是在给定导数时求原函数的过程。如果 F'(x) = f(x)，我们称 F 是 f 的一个原函数。所有原函数的集合称为 f 的不定积分。
Unlike differentiation, integration gives a family of functions rather than one unique result. This is because the derivative of any constant is zero, so many different functions can share the same derivative.
与微分不同，积分给出的是一族函数而不是唯一结果。这是因为任何常数的导数都是零，所以许多不同的函数可以拥有相同的导数。
In Part 1 we focus on indefinite integrals, which do not have limits of integration. Later, definite integrals will be used to calculate exact areas.
在第一部分我们重点讨论不定积分，它不带有积分上下限。后续定积分将用于计算精确面积。
2. Reverse of Differentiation | 微分的逆运算
To integrate f(x), ask: which function F(x) has derivative f(x)? For example, since d/dx (x²) = 2x, an antiderivative of 2x is x².
要对 f(x) 积分，可以问：哪个函数 F(x) 的导数是 f(x)？例如，因为 d/dx (x²) = 2x，所以 2x 的一个原函数是 x²。
However, x² + 3, x² – 5, and x² + 100 all differentiate to 2x. Therefore the general antiderivative is x² + C, where C is any real number.
但是 x² + 3、x² – 5 和 x² + 100 的导数都是 2x。因此一般原函数是 x² + C，其中 C 是任意实数。
∫ 2x dx = x² + C
This shows why the constant C must always be included in an indefinite integral.
这说明为什么不定积分中必须始终加上常数 C。
3. The Constant of Integration | 积分常数
The constant C is called the constant of integration. It represents any real number and covers all possible vertical translations of the antiderivative.
常数 C 称为积分常数。它表示任意实数，涵盖了原函数所有可能的垂直平移。
Geometrically, the graphs of y = F(x) + C are parallel curves because they have the same gradient at every x-value. Adding different values of C simply moves the curve up or down.
从几何上看，y = F(x) + C 的图像是平行曲线，因为它们在每个 x 值处都有相同的斜率。给 C 取不同的值只是把曲线上移或下移。
If a problem gives extra information, such as F(0) = 1, you can substitute into the general solution to find C. This produces a particular solution.
如果题目给了额外信息，例如 F(0) = 1，你可以代入通解求出 C。这样得到的就是特解。
F(x) = ∫ f(x) dx + C
4. Indefinite Integral Notation | 不定积分符号
The notation ∫ f(x) dx means the indefinite integral of f with respect to x. The function f(x) is called the integrand, and dx indicates the variable of integration.
记号 ∫ f(x) dx 表示 f 关于 x 的不定积分。函数 f(x) 称为被积函数，dx 表示积分变量。
Always keep the dx. It is not just decoration; it tells you which letter is the variable. In ∫ 2x dx, the variable is x, not t or y.
必须保留 dx。它不只是装饰，它告诉你哪个字母是变量。在 ∫ 2x dx 中，变量是 x，而不是 t 或 y。
For now, all integrals in this topic are with respect to x, but later you can integrate with respect to other variables using the same rules.
目前本主题中的所有积分都是关于 x 的，但以后你可以对其它变量使用同样的法则进行积分。
5. Power Rule for Integration | 幂函数积分法则
The most important basic rule is the power rule for integration. To integrate xⁿ, add 1 to the power and divide by the new power.
最重要的基本法则是幂函数积分法则。对 xⁿ 积分时，指数加 1，再除以新的指数。
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ -1
This rule works for all real powers except n = -1. The exception n = -1 gives the natural logarithm, which is covered in the standard integrals.
这个法则对所有实数指数都成立，但 n = -1 除外。n = -1 的例外情况对应自然对数，包含在标准积分中。
For example, ∫ x³ dx = x⁴/4 + C, because the new power is 4. The same rule works for negative and fractional powers after rewriting.
例如，∫ x³ dx = x⁴/4 + C，因为新的指数是 4。重写后，该法则对负指数和分数指数同样适用。
6. Integrating Common Functions | 常见函数的积分
Beyond powers of x, you need to know several standard integrals. The table below summarises the most common ones used at this level.
除了 x 的幂，你还需要掌握几个标准积分。下表总结了现阶段最常用的标准积分。
| Function f(x) | Indefinite Integral ∫ 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 |
The absolute value in ln|x| ensures the logarithm is defined for negative x as well. These standard results should be memorised because they are used repeatedly in harder integration problems.
ln|x| 中的绝对值确保对数在 x 为负数时也有定义。这些标准结果需要熟记，因为在更难的积分题中会反复使用。
7. Linearity of Integration | 积分的线性性质
Integration is a linear operation. This means you can integrate term by term and factor out any constant multiplier.
积分是一种线性运算。这意味着你可以逐项积分，并把常数因子提出来。
∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx
For example, ∫ (3x² + 4 cos x) dx = 3 ∫ x² dx + 4 ∫ cos x dx = x³ + 4 sin x + C.
例如，∫ (3x² + 4 cos x) dx = 3 ∫ x² dx + 4 ∫ cos x dx = x³ + 4 sin x + C。
Do not try to integrate a product or quotient directly by integrating each part. Linearity only applies to sums and constant multiples, not to products such as x sin x or quotients such as x/(x + 1).
不要试图通过分别积分各部分来直接处理乘积或商。线性性质只适用于和与常数倍，不适用于 x sin x 这样的乘积或 x/(x + 1) 这样的商。
8. Simplifying Before Integrating | 积分前的化简
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