📚 Integration Part 1: Reversing Differentiation and Indefinite Integrals | 积分第一部分：微分逆运算与不定积分
In A-Level Mathematics, integration is first met as the reverse process of differentiation. Part 1 focuses on indefinite integrals, where we recover a family of functions whose derivative is the given expression. This article develops the notation, the power rule, the constant of integration, and the key skill of finding f(x) from f'(x) that exam papers test again and again.
在 A-Level 数学中，积分最初是作为微分的逆运算引入的。第一部分重点讲解不定积分，即由已知导数恢复出一族原函数。本文将系统整理积分记号、幂函数积分法则、积分常数，以及考试反复考查的由 f'(x) 求 f(x) 的核心技能。
1. What Is Integration? | 什么是积分？
very specific question: given the derivative of a function, what was the original function? If dy/dx = 2x, then y could be x², because d/dx(x²) = 2x. The word 'integrate' literally means 'to make whole again' takes a function apart into its gradient function, and integration puts it back together.
积分是微分的逆运算。它回答的是一个非常具体的问题：已知某个函数的导数，原来的函数是什么？如果 dy/dx = 2x，那么 y 可以是 x²，因为 d/dx(x²) = 2x。'积分'一词的本义就是'重新拼合'：微分把一个函数拆解成它的导函数，积分则把它重新还原。
If F'(x) = f(x), then ∫ f(x) dx = F(x) + C
In symbols, the integral sign ∫ is an elongated letter S, standing for 'sum' e reverse relationship works in both directions: differentiating an integral returns the original integrand, which is exactly how we check our answers later in this article.
用符号表示，积分号 ∫ 是一个拉长的字母 S，表示'求和'，它标示出与 d/dx 相反的操作。写在积分号与 dx 之间的部分，就是要求积分的函数。这种互逆关系是双向的：对一个积分结果求导，会回到原来的被积函数，这正是本文后面检验答案的方法。
Because differentiation is a linear operation, integration is linear as well: constant multiples can be pulled outside the integral sign, and a sum of terms can be integrated term by term. This one property explains why almost every technique in Part 1 works.
由于微分是线性运算，积分同样是线性的：常数因子可以提到积分号外面，和式可以逐项积分。正是这一条性质，使得第一部分中的几乎所有技巧都成立。
∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx
A useful mental model is to think of the derivative as a machine that consumes a function and outputs its gradient, while the integral is the machine that runs backwards, consuming a gradient and outputting a whole family of functions that could have produced it.
一个有用的思维模型是：把求导看作一台机器，输入一个函数、输出它的梯度；而积分则是反向运转的机器，输入一个梯度、输出所有可能产生这个梯度的函数族。
2. Key Notation: The Integral Sign and dx | 关键记号：积分号与 dx
being performed. The integrand is the expression being integrated. The dx tells you which variable you are integrating with respect to, and it closes the integral.
每一个不定积分都由三部分组成，考官期望这三部分都书写正确。积分号 ∫ 告诉读者正在进行什么运算；被积函数是要积分的表达式；dx 则说明对哪个变量积分，同时也标志积分的结束。
| Component | Meaning | 组成部分 | 含义 |
| ∫ | The instruction to integrate | ∫ | 要求积分的指令 |
| f(x) | The integrand, the function to be integrated | f(x) | 被积函数，即要积分的函数 |
| dx | The variable of integration, x | dx | 积分变量为 x |
For example, ∫ (3x² + 1) dx means 'find all functions whose derivative is 3x² + 1, with x as the variable'. The phrase 'with respect to x' with dt, the answer would involve t instead, and the two answers are not interchangeable.
例如，∫ (3x² + 1) dx 的意思是'求出所有导数为 3x² + 1 的函数，变量为 x'。'对 x 积分'这一说法很重要：如果把同一个表达式写成 dt 的形式，答案中出现的将是 t，两者不能混用。
Students often drop the dx under time pressure. Mark schemes in A-Level papers frequently treat a missing dx as a notation error even when the integration itself is correct, so it is worth forming the habit of writing it every single time.
学生在时间压力下常常漏写 dx。A-Level 评分标准往往把漏写 dx 视为记号错误，即使积分本身算对了；因此养成每次都写出 dx 的习惯是值得的。
3. The Constant of Integration | 积分常数 C
Here is the observation that makes integration fundamentally different from differentiation. The functions x² + 1, x² − 3 and x² + 100 all differentiate to 2x, because the derivative of any constant is zero. Integration therefore gives a family of functions, not just one function.
正是下面这个观察使积分与微分有本质区别。函数 x² + 1、x² − 3、x² + 100 的导数都是 2x，因为任何常数的导数都是零。因此积分得到的是一个函数族，而不是唯一的一个函数。
∫ 2x dx = x² + C
The letter C denotes an arbitrary constant, and it must be included in every indefinite integral. Geometrically, adding C translates the curve vertically. Each value of C gives a curve with exactly the same shape and the same gradient at every value of x, but at a different height on the page.
字母 C 表示任意常数，每一个不定积分都必须写上它。从几何上看，加上 C 相当于把曲线上下平移。每一个 C 值给出的曲线形状完全相同，在任意 x 处的梯度也完全相同，只是在图上的高度不同。
Some exam questions ask for 'the general solution' of a differential equation. In that case, an answer without +C loses the mark for the general constant, even if the algebraic part is perfect. Only when an extra condition (a point on the curve, or a value of f(x)) is supplied does C become a specific number, and we then say we have found a 'particular solution'.
有些考题要求求微分方程的'通解'。这时如果答案没有写 +C，即使代数部分完全正确，也会丢掉表示任意常数的那一分。只有当题目额外给出条件（曲线上的一点，或 f(x) 的某个值）时，C 才变成确定数值，这时我们得到的称为'特解'。
4. The Power Rule for Integration | 幂函数积分法则
The power rule for integration is the reverse of the power rule for differentiation, and it is the single most used result in this topic. To integrate a power of x, raise the power by one and divide by the new power.
幂函数积分法则是幂函数求导法则的逆过程，也是本专题中使用频率最高的结论。对 x 的幂函数积分，先把指数加一，再除以新的指数。
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, for n ≠ −1
sion by zero, so x⁻¹ must be handled separately; that case is dealt with later in this article. For every other real value of n, including negative and fractional values, the rule works exactly as stated.
条件 n ≠ −1 至关重要。如果 n = −1，公式就会出现除以零，因此 x⁻¹ 必须单独处理，这一情形本文后面会讨论。对于其他任何实数 n，包括负数和分数，该法则都完全适用。
x⁵, which is exactly what we started with. This two-second check should become automatic.
例如，∫ x⁵ dx = x⁶ / 6 + C。我们可以立刻通过求导验证：d/dx(x⁶ / 6) = 6x⁵ / 6 = x⁵，正好回到原来的被积函数。这个只需两秒的检验应该成为条件反射。
A frequent slip is to subtract one from the power, which is what we do when differentiating. Remember the direction: differentiating lowers the power, integrating raises it. Writing 'raise the power, divide by the new power' above your working is a cheap way to avoid this error.
一个常见失误是把指数减一，那是求导的做法。要记住方向：求导使次数降低，积分使次数升高。在解题过程上方写上'指数加一，除以新指数'，是避免这个错误的低成本办法。
5. Integrating Polynomials Term by Term | 逐项积分多项式
he results. Constant terms are written as kx⁰ and integrated to kx, since ∫ k dx = kx + C.
由于积分具有线性性质，多项式的积分就是对每一项分别使用幂函数法则，再把结果相加。常数项可以写成 kx⁰，积分后得到 kx，因为 ∫ k dx = kx + C。
∫ (6x² − 4x + 3) dx = 2x³ − 2x² + 3x + C
stant 3 integrates to 3x. Although each separate integral would produce its own constant, we combine them into one single C at the end, since a sum of arbitrary constants is still just an arbitrary constant.
逐项看这个计算：6x² 积分得 6x³/3 = 2x³；−4x 积分得 −4x²/2 = −2x²；常数 3 积分得 3x。虽然每一项单独积分都会产生一个常数，但最后要合并成一个 C，因为任意常数之和仍然是任意常数。
integrates to 2x. Notice how the division by the new power cancels the coefficient cleanly whenever the original coefficient is a multiple of the new power - a pattern that appears constantly in exam questions.
再看一个带系数的例子：∫ (4x³ + 2) dx = x⁴ + 2x + C。这里 4x³ 积分得 4x⁴/4 = x⁴，常数 2 积分得 2x。注意当原系数恰好是新指数的倍数时，除以新指数会把系数干净地约掉，这一模式在考题中反复出现。
6. Negative and Fractional Powers | 负指数与分数指数
Examiners love to disguise simple integrals as roots and reciprocals. The method never changes: rewrite every term as a power of x first, then apply the power rule. A root such as √x becomes x^½, and a reciprocal such as 1/x³ becomes x⁻³.
考官很喜欢把简单的积分伪装成根式和倒数。方法始终不变：先把每一项改写为 x 的幂，再套用幂函数法则。像 √x 这样的根式写成 x^½，像 1/x³ 这样的倒数写成 x⁻³。
∫ x⁻³ dx = x⁻² / (−2) + C = −1 / (2x²) + C
∫ √x dx = ∫ x^½ dx = (2/3)x3/2 + C
In the second example, dividing by 3/2 is the same as multiplying by 2/3, which is where the fraction (2/3) comes from. Fraction arithmetic is the most common source of lost marks here, so it is worth writing the division as a separate step before simplifying.
在第二个例子中，除以 3/2 等同于乘以 2/3，这就是分数 (2/3) 的来源。分数运算是这里失分最多的地方，因此值得把除法单独写成一步再化简。
Two more patterns worth memorising: ∫ 1/x² dx = ∫ x⁻² dx = −x⁻¹ + C = −1/x + C, and ∫ 1/√x dx = ∫ x−½ dx = 2x^½ + C = 2√x + C. Notice that the answer to the second one is a root rather than a fraction, which is a good sanity check that the arithmetic has not gone wrong.
还有两个值得记牢的模式：∫ 1/x² dx = ∫ x⁻² dx = −x⁻¹ + C = −1/x + C，以及 ∫ 1/√x dx = ∫ x−½ dx = 2x^½ + C = 2√x + C。注意第二个的答案是根式而不是分式，这是检验运算是否出错的一个好办法。
2, so the integral is x³/3 + 2x + C. Trying to integrate a quotient directly, or reaching for a quotient rule, is never the right move at this level.
当被积函数是像 (x³ + 2x)/x 这样的商时，一定先化简再积分：(x³ + 2x)/x = x² + 2，所以积分是 x³/3 + 2x + C。在这个阶段，直接对商积分或者想套商的法则，永远不是正确做法。
7. Always Check by Differentiating | 用求导检验答案
Every indefinite integral you write can be verified in one line, because differentiation undoes integration. Differentiate your answer, ignore the constant C, and see whether you recover the original integrand. If you do, the answer is correct; if you do not, the error is somewhere in the powers or the coefficients.
你写出的每一个不定积分都可以用一行验证，因为求导可以抵消积分。对答案求导，忽略常数 C，看是否回到原来的被积函数。如果回到了，答案就正确；如果没有，错误一定出在指数或系数上。
∫ (2x + 3) dx = x² + 3x + C, check: d/dx(x² + 3x) = 2x + 3 ✓
wrote −1/(2x²) + C for ∫ x⁻³ dx, differentiating gives d/dx(−1/(2x²)) = d/dx(−½ x⁻²) = (−½)(−2)x⁻³ = x⁻³, confirming the answer.
对于负指数和分数指数，这个检验尤其宝贵，因为正负号和分数都很容易出错。例如，如果你对 ∫ x⁻³ dx 写出 −1/(2x²) + C，求导得 d/dx(−1/(2x²)) = d/dx(−½ x⁻²) = (−½)(−2)x⁻³ = x⁻³，正好验证了答案。
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