A-Level Maths: Indefinite Integrals – Basic Concepts & Computation | A-Level数学：不定积分的基本概念与计算

📚 A-Level Maths: Indefinite Integrals – Basic Concepts & Computation | A-Level数学：不定积分的基本概念与计算

Integration is one of the two core operations in calculus, alongside differentiation. In A-Level Mathematics, indefinite integration plays a pivotal role, and mastering its basic concepts and computational techniques is essential for success in the Edexcel examinations.

积分是微积分中与微分并列的两大核心运算之一。在A-Level数学中，不定积分占有举足轻重的地位，掌握其基本概念与计算方法对于在Edexcel考试中取得好成绩至关重要。

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1. What Is Integration? | 什么是积分？

Integration is the reverse process of differentiation. If differentiating a function F(x) gives f(x), then integrating f(x) returns the family of functions that include F(x). We call this process finding the antiderivative, or the indefinite integral.

积分是微分的逆运算。如果对函数 F(x) 求导得到 f(x)，那么对 f(x) 进行积分就能还原出包含 F(x) 在内的函数族。这一过程称为求原函数或不定积分。

For example, since the derivative of x² is 2x, the indefinite integral of 2x with respect to x is x² + C.

例如，由于 x² 的导数是 2x，因此 2x 关于 x 的不定积分是 x² + C。

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2. Notation and the Constant of Integration | 记号与积分常数

The indefinite integral of a function f(x) with respect to x is written as:

已知函数 f(x) 关于变量 x 的不定积分记作：

∫ f(x) dx = F(x) + C

Here, ∫ is the integral sign, dx indicates that we integrate with respect to x, F(x) is any antiderivative of f(x), and C is the constant of integration.

其中，∫ 是积分号，dx 表示对变量 x 积分，F(x) 是 f(x) 的任意一个原函数，C 称为积分常数。

Because the derivative of any constant is zero, adding C accounts for all possible vertical shifts of the antiderivative. Therefore, an indefinite integral always represents a family of curves, not a single curve.

由于任意常数的导数均为零，加上 C 可以涵盖原函数的所有可能竖直平移。因此，不定积分始终表示一族曲线，而非一条唯一的曲线。

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3. The Power Rule | 幂函数积分法则

The most fundamental rule in integration is the power rule, which states that for any real number n ≠ -1:

积分中最基本的方法是幂函数法则，它指出：对任意实数 n ≠ -1，有：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

To apply the rule, add 1 to the exponent and divide by the new exponent, then add the constant C.

运用该法则时，先将指数加 1，再除以新的指数，最后加上常数 C。

For example, ∫ x³ dx = x⁴/4 + C. Check: differentiating x⁴/4 gives 4x³/4 = x³. ✓

例如，∫ x³ dx = x⁴/4 + C。验证：对 x⁴/4 求导得到 4x³/4 = x³。✓

Special case: when n = 0, the rule gives ∫ 1 dx = x + C, which is the integral of a constant.

特殊情况：当 n = 0 时，该法则给出 ∫ 1 dx = x + C，即常数的积分。

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4. Integration of Standard Functions | 基本函数积分表

Beyond the power rule, A-Level students must memorise the integrals of standard functions. The table below summarises the most important ones:

除幂函数法则外，A-Level 学生还需牢记基本函数的积分公式。下表总结了最常用的几项：

f(x)	∫ f(x) dx
xⁿ (n ≠ -1)	xⁿ⁺¹/(n+1) + C
1/x (x ≠ 0)	ln|x| + C
eˣ	eˣ + C
sin x	-cos x + C
cos x	sin x + C
sec² x	tan x + C

Note that the integral of 1/x is a special case of the power rule that must be handled separately, because the formula xⁿ⁺¹/(n+1) is undefined when n = -1.

注意：1/x 的积分是幂法则的特例，必须单独处理，因为当 n = -1 时公式 xⁿ⁺¹/(n+1) 无定义。

For instance, ∫ 4/x dx = 4 ln|x| + C, and ∫ (2eˣ + cos x) dx = 2eˣ + sin x + C.

例如，∫ 4/x dx = 4 ln|x| + C，而 ∫ (2eˣ + cos x) dx = 2eˣ + sin x + C。

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5. Rules: Constant Multiple & Sum/Difference | 积分运算法则

Two linearity rules make integration far more manageable when dealing with combinations of functions.

两条线性运算法则可以帮助我们轻松处理函数的组合。

Rule 1 — Constant Multiple: ∫ k·f(x) dx = k·∫ f(x) dx, where k is a constant.

法则一——常数倍： ∫ k·f(x) dx = k·∫ f(x) dx，其中 k 为常数。

This rule allows us to pull any constant factor outside the integral sign.

该法则允许我们将任何常数因子提到积分号外面。

Rule 2 — Sum/Difference: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.

法则二——和差： ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ±

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