Indefinite Integrals | 不定积分

📚 Indefinite Integrals | 不定积分

Indefinite integration is one of the two main branches of calculus, the reverse process of differentiation. In A-Level Edexcel Mathematics, you learn how to recover a function from its derivative, producing a family of functions that all differ by a constant. Mastering indefinite integrals is essential for solving differential equations, finding areas, and tackling applied problems across pure and mechanics modules.

不定积分是微积分的两大分支之一，也就是微分的逆运算。在Edexcel A-Level数学中，你需要学会如何由导函数还原出原函数，得到的是一个相差常数的函数族。掌握不定积分对求解微分方程、计算面积以及处理纯数学和力学中的应用问题至关重要。

1. What is an Indefinite Integral? | 不定积分的概念

An indefinite integral of a function f(x) is a function F(x) whose derivative is f(x). We write it as ∫ f(x) dx = F(x) + C gn, f(x) is the integrand, and dx indicates the variable of integration. The process of finding F is called antidifferentiation.

函数 f(x) 的不定积分是指导数等于 f(x) 的函数 F(x)。我们记作 ∫ f(x) dx = F(x) + C，其中 ∫ 是积分号，f(x) 是被积函数，dx 是积分变量。求出 F 的过程称为反微分。

Because the derivative of any constant is zero, there are infinitely many antiderivatives differing only by a constant. The notation emphasises that integration gives a family of curves rather than a single function.

由于任何常数的导数都为零，因此有无穷多个原函数，它们只差一个常数。这个记法强调了积分给出的是一族曲线，而不是一个单独的函数。

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2. The Constant of Integration | 积分常数C

Whenever you evaluate an indefinite integral, you must include the constant of integration, typically denoted by C. For example, ∫ 2x dx = x² + C because the derivative of x² + C is 2x regardless of the value of C.

每当你计算不定积分时，都必须加上积分常数，通常用 C 表示。例如，∫ 2x dx = x² + C，因为无论 C 取何值，x² + C 的导数都是 2x。

Omitting C on an A-Level exam paper will cost you marks, especially in questions that ask for the general solution of a differential equation or require you to find a particular solution using given conditions.

在A-Level考试中遗漏 C 会丢分，尤其是在求微分方程的通解或者需要利用给定条件求出特解的题目中。

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3. Basic Integration Rules: The Power Rule | 基本幂函数积分法则

The most fundamental tool is the power rule for integration. For any real number n ≠ –1,

∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C.

最基本的工具是幂函数的积分法则。对于任意实数 n ≠ –1，有

∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C。

This rule works for positive integers, negative powers, and fractional indices. For instance, ∫ x³ dx = x⁴/4 + C, ∫ 1/x² dx = ∫ x⁻² dx = –x⁻¹ + C = –1/x + C, and ∫ √x dx = ∫ x½ dx = (2/3) x³/² + C. Always rewrite roots and reciprocals as powers before integrating.

该法则适用于正整数指数、负指数和分数指数。例如，∫ x³ dx = x⁴/4 + C，∫ 1/x² dx = ∫ x⁻² dx = –x⁻¹ + C = –1/x + C，以及 ∫ √x dx = ∫ x½ dx = (2/3) x³/² + C。在积分前务必把根式和倒数改写为幂的形式。

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4. Integrating Exponential Functions | 指数函数的积分

The exponential function e^x is special because it is its own derivative and antiderivative. The basic rule is

∫ e^x dx = e^x + C.

指数函数 e^x 很特别，因为它既是自身的导数也是自身的原函数。基本法则为

∫ e^x dx = e^x + C。

More generally, for a constant a > 0, a ≠ 1, we have ∫ a^x dx = a^x / ln a + C. This formula can be derived by writing a^x = e^{x ln a} and using the reverse chain rule.

更一般地，对于常数 a > 0 且 a ≠ 1，有 ∫ a^x dx = a^x / ln a + C。这个公式可以通过将 a^x 写成 e^{x ln a} 再利用逆链式法则推得。

When the exponent is a linear function, e.g., ∫ e^kx dx, you must divide by the coefficient of x: ∫ e^kx dx = (1/k) e^kx + C.

当指数是线性函数时，例如 ∫ e^kx dx，需要除以 x 的系数：∫ e^kx dx = (1/k) e^kx + C。

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5. Integrating Trigonometric Functions | 三角函数的积分

A-Level candidates must memorise the standard integrals of the core trigonometric functions. The derivatives give the corresponding integrals directly:

• ∫ sin x dx = –cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C
• ∫ cosec x cot x dx = –cosec x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec² x dx = –cot x + C

A-Level考生必须熟记核心三角函数的积分标准公式。由导数直接得出对应的积分：

• ∫ sin x dx = –cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C
• ∫ cosec x cot x dx = –cosec x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec² x dx = –cot x + C

gns; a common mistake is to write ∫ sin x dx = cos x + C, which is missing the minus sign.

务必注意符号；一种常见错误是把 ∫ sin x dx 写成了 cos x + C，遗漏了负号。

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6. Integrating 1/x and the Natural Logarithm | 1/x 与自然对数的积分

The power rule fails when n = –1 because division by zero is undefined. Instead, we use

∫ 1/x dx = ln|x| + C.

当 n = –1 时幂法则不适用，因为分母为零。此时，我们使用

∫ 1/x dx = ln|x| + C。

able for negative x as well. This integral often appears as part of a broader pattern: ∫ f'(x)/f(x) dx.

这里必须要有绝对值，因为自然对数只对正数有意义，而 1/x 在 x 为负时也可积。这个积分经常作为更广的形式 ∫ f'(x)/f(x) dx 的一部分出现。

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7. Reverse Chain Rule with Linear Inner Functions | 逆链式法则（线性内函数）

When the integrand is a function of a linear expression ax + b, you can integrate by "reversing" the chain rule. In general,

∫ f'(ax + b) dx = (1/a) f(ax + b) + C.

当被积函数是线性表达式 ax + b 的函数时，可以通过“反向”链式法则积分。一般地，

∫ f'(ax + b) dx = (1/a) f(ax + b) + C。

Examples: ∫ (2x + 3)⁴ dx = (1/2) × (1/5)(2x + 3)⁵ + C = (1/10)(2x + 3)⁵ + C; ∫ sin(4x) dx = –(1/4) cos(4x) + C; ∫ e^3x dx = (1/3) e^3x + C. This technique saves time and avoids formal substitution.

例如：∫ (2x + 3)⁴ dx = (1/2) × (1/5)(2x + 3)⁵ + C = (1/10)(2x + 3)⁵ + C；∫ sin(4x) dx = –(1/4) cos(4x) + C；∫ e^3x dx = (1/3) e^3x + C。这种技巧可以节省时间，避免形式的换元。

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8. The f'(x)/f(x) Form | f'(x)/f(x) 类型的积分

When the numerator is the exact derivative of the denominator, the integral simplifies to a natural logarithm:

∫ f'(x)/f(x) dx = ln|f(x)| + C.

当分子恰好是分母的导数时，积分可化为自然对数：

∫ f'(x)/f(x) dx = ln|f(x)| + C。

For example, ∫ (2x)/(x² + 5) dx = ln|x² + 5| + C because the derivative of x² + 5 is 2x. If the numerator is a constant multiple of the derivative, factor the constant out first: ∫ x/(x² + 1) dx = ½ ∫ 2x/(x² + 1) dx = ½ ln(x² + 1) + C.

例如，∫ (2x)/(x² + 5) dx = ln|x² + 5| + C，因为 x² + 5 的导数是 2x。如果分子是导数的常数倍，则先把常数提出来：∫ x/(x² + 1) dx = ½ ∫ 2x/(x² + 1) dx = ½ ln(x² + 1) + C。

Spotting this pattern is a highly examinable skill in Edexcel Pure Mathematics. Always check whether the integrand can be manipulated into the f'(x)/f(x) form.

识别这种模式是Edexcel纯数学中极易考查的技能。要经常检查被积函数是否可以转化为 f'(x)/f(x) 的形式。

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9. Using Trigonometric Identities | 利用三角恒等式积分

Many trigonometric integrals require identities to rewrite the integrand into a form that matches standard results. For powers of sin and cos, the double-angle identities are crucial.

许多三角积分需要借助恒等式将被积函数改写成符合标准公式的形式。对于 sin 和 cos 的幂，倍角公式至关重要。

Examples: ∫ sin²x dx. Use cos 2x = 1 – 2 sin²x ⇒ sin²x = ½(1 – cos 2x). Then ∫ sin²x dx = ½ ∫ (1 – cos 2x) dx = ½ (x – ½ sin 2x) + C = ½ x – ¼ sin 2x + C.

例如：∫ sin²x dx。利用 cos 2x = 1 – 2 sin²x ⇒ sin²x = ½(1 – cos 2x)。则 ∫ sin²x dx = ½ ∫ (1 – cos 2x) dx = ½ (x – ½ sin 2x) + C = ½ x – ¼ sin 2x + C。

Similarly, ∫ cos²x dx uses cos²x = ½(1 + cos 2x), and ∫ tan²x dx = ∫ (sec²x – 1) dx = tan x – x + C. For products such as sin 3x cos x, apply the product-to-sum formulas.

类似地，∫ cos²x dx 使用 cos²x = ½(1 + cos 2x)，而 ∫ tan²x dx = ∫ (sec²x – 1) dx = tan x – x + C。对于乘积如 sin 3x cos x，则采用积化和差公式。

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10. Integration by Substitution | 换元积分法

on (often called u-substitution) systematically transforms the integral. Given ∫ f(g(x)) g'(x) dx, let u = g(x), then du/dx = g'(x) so that du = g'(x) dx, and the integral becomes ∫ f(u) du.

当逆链式法则不那么直接时，换元法（常称为 u 代换）可以系统性地转化积分。对于 ∫ f(g(x)) g'(x) dx，设 u = g(x)，则 du/dx = g'(x)，从而 du = g'(x) dx，积分化为 ∫ f(u) du。

Example: ∫ x (x² + 1)⁴ dx. Let u = x² + 1, so du = 2x dx ⇒ x dx = ½ du. The integral becomes ∫ u⁴ · ½ du = ½ × u⁵/5 + C = (1/10)(x² + 1)⁵ + C. Always remember to replace u by the original variable in your final answer.

例如：∫ x (x² + 1)⁴ dx。设 u = x² + 1，则 du = 2x dx ⇒ x dx = ½ du。积分变为 ∫ u⁴ · ½ du = ½ × u⁵/5 + C = (1/10)(x² + 1)⁵ + C。务必在最终答案中将 u 换回原变量。

For definite integrals the limits must also be changed, but for indefinite integrals the final expression must be in terms of x.

定积分还需改变上下限，但对于不定积分，最终表达式必须用 x 表示。

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11. Integration by Parts | 分部积分法

Integration by parts is used when the integrand is a product of two functions and substitution does not help. The formula, derived from the product rule, is

∫ u (dv/dx) dx = u v − ∫ v (du/dx) dx.

当被积函数是两个函数的乘积且换元法无效时，使用分部积分法。该公式由乘法法则导出，

∫ u (dv/dx) dx = u v − ∫ v (du/dx) dx。

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