Integration of Basic Functions: A Complete Formula Guide | A-Level 数学：基本函数积分公式整合

📚 Integration of Basic Functions: A Complete Formula Guide | A-Level 数学：基本函数积分公式整合

Integration is one of the two central operations in calculus. In this guide, we consolidate all the basic function integration formulas you need for A-Level Mathematics, explain how to apply them, and highlight common pitfalls.

积分是微积分两大核心运算之一。本指南整合了 A-Level 数学中所有基本函数的积分公式，讲解其应用方法，并指出常见易错点。

1. The Power Rule | 幂函数积分法则

The most fundamental integration rule states that for any constant n ≠ -1, the integral of xⁿ is xⁿ⁺¹/(n+1), plus a constant of integration C. This rule applies to all real values of n except -1.

最基本的积分法则指出：对于任意常数 n ≠ -1，xⁿ 的积分为 xⁿ⁺¹/(n+1)，再加上积分常数 C。该法则适用于除 -1 外的所有实数 n。

If n = 0, then ∫ x⁰ dx = ∫ 1 dx = x + C.
If n = 1, then ∫ x dx = x²/2 + C.
If n = -2, then ∫ x⁻² dx = -x⁻¹ + C = -1/x + C.

当 n = 0 时，∫ x⁰ dx = ∫ 1 dx = x + C。
当 n = 1 时，∫ x dx = x²/2 + C。
当 n = -2 时，∫ x⁻² dx = -x⁻¹ + C = -1/x + C。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ -1）

2. The Natural Logarithm Rule | 自然对数法则

The single exception to the power rule is n = -1. The integral of x⁻¹ is the natural logarithm of the absolute value of x. This absolute value is essential because the logarithm is only defined for positive arguments, but the function 1/x is defined for both positive and negative x.

幂法则的唯一例外是 n = -1。x⁻¹ 的积分是 x 绝对值的自然对数。绝对值必不可少，因为对数只对正数有定义，而 1/x 在正数和负数上都有定义。

∫ x⁻¹ dx = ∫ 1/x dx = ln|x| + C

When the integrand is of the form f'(x)/f(x), the integral is ln|f(x)| + C. For example, ∫ (2x)/(x²+1) dx = ln(x²+1) + C.

当被积函数形如 f'(x)/f(x) 时，其积分为 ln|f(x)| + C。例如，∫ (2x)/(x²+1) dx = ln(x²+1) + C。

3. Exponential Functions | 指数函数积分

The exponential function eˣ is unique: its integral is itself. More generally, for a constant base a > 0, a ≠ 1, the integral of aˣ is aˣ / ln a + C. A common special case is ∫ e^(kx) dx = (1/k)e^(kx) + C.

指数函数 eˣ 独一无二：其积分等于自身。更一般地，对于常数底数 a > 0，a ≠ 1，aˣ 的积分为 aˣ / ln a + C。常见特例是 ∫ e^(kx) dx = (1/k)e^(kx) + C。

∫ eˣ dx = eˣ + C, ∫ aˣ dx = aˣ/ln a + C, ∫ e^(kx) dx = (1/k)e^(kx) + C

∫ eˣ dx = eˣ + C，∫ aˣ dx = aˣ/ln a + C，∫ e^(kx) dx = (1/k)e^(kx) + C

Remember to divide by the derivative of the exponent when using reverse chain rule for e^(kx).

使用反链式法则计算 e^(kx) 时，记得除以指数部分的导数。

4. Trigonometric Functions | 三角函数积分

The integrals of the six basic trigonometric functions are derived from differentiation in reverse. Sin and cos are straightforward; tan, cot, sec and csc require logarithmic forms.

六个基本三角函数的积分由微分反向推导得出。sin 和 cos 直接；tan、cot、sec 和 csc 需要对数形式。

∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C, ∫ tan x dx = ln|sec x| + C

∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C，∫ tan x dx = ln|sec x| + C

For linear arguments like sin(ax+b), divide by the coefficient a.

对于线性参数如 sin(ax+b)，需除以系数 a。

5. Inverse Trigonometric Functions | 反三角函数积分

A-Level further mathematics often requires the integrals of 1/√(a² – x²) and 1/(a² + x²). These yield inverse sine and inverse tangent functions respectively.

A-Level 进阶数学常要求积分 1/√(a² – x²) 和 1/(a² + x²)。它们分别得到反正弦和反正切函数。

∫ 1/√(a² – x²) dx = arcsin(x/a) + C, ∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C

∫ 1/√(a² – x²) dx = arcsin(x/a) + C，∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C

These formulas are particularly useful when completing the square appears in rational expressions.

这些公式在处理含有配方的有理表达式时特别有用。

6. Linearity of Integration | 积分的线性性质

Integration is linear: the integral of a sum is the sum of integrals, and constant factors can be pulled out. This property allows us to integrate polynomials term by term.

积分具有线性：和的积分等于积分的和，常数因子可以提出。这一性质允许我们逐项积分多项式。

∫ [af(x) + bg(x)] dx = a∫ f(x) dx + b∫ g(x) dx

Example: ∫ (3x² + 4x – 5) dx = x³ + 2x² – 5x + C.

示例：∫ (3x² + 4x – 5) dx = x³ + 2x² – 5x + C。

7. Integration by Substitution | 换元积分法

Substitution (reverse chain rule) handles composite functions. If we set u = g(x), then dx is replaced by du/g'(x). The goal is to transform the integral into a simpler one in terms of u.

换元积分法（反链式法则）处理复合函数。令 u = g(x)，则 dx 替换为 du/g'(x)。目标是将积分转化为关于 u 的更简单形式。

∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x)

∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)

A typical A-Level exam example: ∫ x e^(x²) dx. Let u = x², du = 2x dx, so the integral becomes (1/2)∫ eᵘ du = (1/2)e^(x²) + C.

典型 A-Level 考题：∫ x e^(x²) dx。令 u = x²，du = 2x dx，则积分变为 (1/2)∫ eᵘ du = (1/2)e^(x²) + C。

8. Integration by Parts | 分部积分法

Integration by parts is derived from the product rule and is used when the integrand is a product of two different types of functions (e.g., polynomial × exponential, or logarithmic × polynomial).

分部积分法由乘积法则推导而来，用于被积函数是两类不同函数乘积的情形（例如多项式×指数，或对数×多项式）。

∫ u dv = uv – ∫ v du

For example, ∫ x cos x dx. Choose u = x, dv = cos x dx, then du = dx and v = sin x. The result is x sin x – ∫ sin x dx = x sin x + cos x + C.

例如，∫ x cos x dx。选 u = x，dv = cos x dx，则 du = dx，v = sin x。结果为 x sin x – ∫ sin x dx = x sin x + cos x + C。

9. Definite Integrals and Areas | 定积分与面积

A definite integral computes the net signed area between the curve and the x-axis over an interval [a, b]. We evaluate the antiderivative at b and subtract its value at a.

定积分计算区间 [a, b] 上曲线与 x 轴之间的净符号面积。我们求原函数在 b 处的值并减去在 a 处的值。

∫ₐᵇ f(x) dx = F(b) – F(a), where F'(x) = f(x)

∫ₐᵇ f(x) dx = F(b) – F(a)，其中 F'(x) = f(x)

When the curve lies below the x-axis, the definite integral is negative. To find total area, integrate the absolute value or split the interval at roots.

当曲线位于 x 轴下方时，定积分为负。要求总面积，需对绝对值积分或在根处分割区间。

10. Common Pitfalls and Exam Tips | 常见易错点与考试技巧

Students often forget the constant of integration in indefinite integrals, miss the absolute value in ln|x|, or fail to handle negative powers correctly.

学生常忘记不定积分中的积分常数，遗漏 ln|x| 中的绝对值，或未能正确处理负幂次。

Always add + C for indefinite integrals.
For ∫ (ax + b)ⁿ dx with n ≠ -1, use (ax + b)ⁿ⁺¹ / [a(n+1)].
When using substitution in definite integrals, change the limits of integration.
Check your answer by differentiating it.

不定积分永远加 + C。
对于 ∫ (ax + b)ⁿ dx（n ≠ -1），使用 (ax + b)ⁿ⁺¹ / [a(n+1)]。
在定积分中使用换元法时，要更换积分上下限。
通过求导检验你的答案。

11. Quick Reference Table | 快速公式表

The table below summarises the most essential integrals for A-Level Mathematics.

下表总结了 A-Level 数学最核心的积分公式。

Function	Integral
xⁿ (n ≠ -1)	xⁿ⁺¹/(n+1) + C
1/x	ln\|x\| + C
eᵏˣ	(1/k)eᵏˣ + C
cos x	sin x + C
sin x	-cos x + C
sec² x	tan x + C
1/√(a² – x²)	arcsin(x/a) + C
1/(a² + x²)	(1/a) arctan(x/a) + C

Mastering these formulas and their derivations will build a solid foundation for solving integration problems in both pure and applied mathematics.

掌握这些公式及其推导，将为你在纯数学和应用数学中解决积分问题奠定坚实基础。

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Integration of Basic Functions: A Complete Formula Guide | A-Level 数学：基本函数积分公式整合