Common Standard Function Integrals | 常见标准函数的积分公式

📚 Common Standard Function Integrals | 常见标准函数的积分公式

In A-Level mathematics, mastering the integration of standard functions is essential for solving problems in calculus, physics, and further mathematics. This guide covers the most frequently tested formulas, with clear explanations and worked examples to help you apply them with confidence.

在 A-Level 数学中，掌握常见标准函数的积分公式是解决微积分、物理及进阶数学问题的关键。本篇指南涵盖考试中最常考的积分公式，并配有清晰的讲解和例题，帮助你自信地运用它们。

1. Power Rule for xⁿ | xⁿ 的幂函数积分公式

The power rule states that the integral of xⁿ, where n is a constant and n ≠ -1, is given by xⁿ⁺¹ divided by n+1, plus a constant of integration C. This rule is the reverse of differentiation for simple polynomials.

幂函数积分公式指出：当 n 为常数且 n ≠ -1 时，xⁿ 的积分为 xⁿ⁺¹ 除以 n+1，再加上积分常数 C。该公式是简单多项式求导的逆运算。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1

For example, the integral of x² is x³/3 + C. To verify, differentiate x³/3 and you obtain x².

例如，x² 的积分为 x³/3 + C。可通过求导 x³/3 来验证，结果确实是 x²。

When integrating polynomials, apply the rule term by term. Constants can be moved outside the integral sign.

对多项式积分时，逐项应用公式即可。常数可以提到积分号外面。

Rule valid for all real n except n = -1.
Use indefinite integrals with + C.
For definite integrals, evaluate at limits and subtract.

该公式适用于除 n = -1 以外的所有实数 n。
不定积分要注意加上常数 C。
定积分则在上下限处取值并相减。

2. The Special Case: 1/x | 特殊情形：1/x

The function 1/x, or x⁻¹, cannot be integrated using the power rule because it would lead to division by zero. Instead, its integral is the natural logarithm of the absolute value of x, plus C.

函数 1/x（即 x⁻¹）不能使用幂函数积分公式，因为会导致分母为零。它的积分是自然对数绝对值 |x|，再加上常数 C。

∫ (1/x) dx = ln|x| + C

The absolute value ensures the logarithm is defined for negative x as well. In many exam problems, you may see the notation ln x when x is positive.

绝对值符号确保对负 x 时对数也有定义。在考试中，当 x 为正时有时直接写成 ln x。

This formula appears frequently in problems involving exponential growth, velocity, and areas under hyperbolas.

这个公式常出现在涉及指数增长、速度以及双曲线下面积的问题中。

Remember the modulus sign in ln|x|.
Use base e (natural logarithm) unless specified.
Related formula: ∫ (f'(x)/f(x)) dx = ln|f(x)| + C.

记住 ln|x| 中的绝对值符号。
使用自然对数（以 e 为底），除非题目另有说明。
相关公式：∫ (f'(x)/f(x)) dx = ln|f(x)| + C。

3. Exponential Functions eˣ and eᵏˣ | 指数函数 eˣ 与 eᵏˣ

The exponential function eˣ is the only function that equals its own derivative and its own integral. For a general exponential eᵏˣ, the integral divides by the constant k.

指数函数 eˣ 是唯一与其导数及其积分相等的函数。对一般形式 eᵏˣ，积分时除以常数 k 即可。

∫ eˣ dx = eˣ + C; ∫ eᵏˣ dx = eᵏˣ/k + C, k ≠ 0

For instance, the integral of e³ˣ is e³ˣ/3 + C. This can be verified by differentiation using the chain rule.

例如，e³ˣ 的积分为 e³ˣ/3 + C。用链式法则求导即可验证。

When the exponent is a linear function ax + b, the integral is e^(ax+b) divided by a, plus C.

当指数是线性函数 ax + b 时，其积分为 e^(ax+b) 除以 a，再加上 C。

Never add extra terms when integrating eˣ.
Check if the exponent is linear before applying the rule directly.
Use substitution for non-linear exponents.

对 eˣ 积分时不要添加额外项。
先确认指数是否为线性函数，再直接套用公式。
非线性的指数需要使用换元法处理。

4. Trigonometric Functions sin x and cos x | 三角函数 sin x 与 cos x

The integrals of the basic trig functions follow directly from differentiation rules. The integral of sin x is -cos x + C, and the integral of cos x is sin x + C.

基本三角函数的积分可以直接由求导法则得出。sin x 的积分是 -cos x + C，cos x 的积分是 sin x + C。

∫ sin x dx = -cos x + C; ∫ cos x dx = sin x + C

For linear arguments like sin(kx) or cos(kx), divide by k. For example, the integral of sin(2x) is -cos(2x)/2 + C.

对于线性参数如 sin(kx) 或 cos(kx)，需要除以 k。例如，sin(2x) 的积分是 -cos(2x)/2 + C。

These formulas are the foundation for integrating more complex trigonometric expressions using identities and substitution.

这些公式是利用恒等式和换元法处理更复杂三角表达式的积分基础。

Memorise the sign: integral of sin involves a negative sign.
General form: ∫ sin(kx) dx = -cos(kx)/k + C.
Always add + C for indefinite integrals.

记住正负号：sin 的积分带负号。
一般形式：∫ sin(kx) dx = -cos(kx)/k + C。
不定积分别忘了加 + C。

5. Sec² x and Standard Trig Derivatives | sec² x 与标准三角函数导数

The integral of sec² x is tan x + C. This is because the derivative of tan x is sec² x. Similarly, the integral of sec x tan x is sec x + C.

sec² x 的积分是 tan x + C，因为 tan x 的导数是 sec² x。类似地，sec x tan x 的积分是 sec x + C。

∫ sec² x dx = tan x + C; ∫ sec x tan x dx = sec x + C

These two results often appear in integration by substitution and in problems involving trigonometry after applying identities.

这两个结果常出现在换元积分法中，以及需要先用恒等式化简三角函数的题目里。

Also note that ∫ cosec² x dx = -cot x + C and ∫ cosec x cot x dx = -cosec x + C, though these are less common in A-Level.

另外注意 ∫ csc² x dx = -cot x + C，以及 ∫ csc x cot x dx = -csc x + C，不过在 A-Level 中不常出现。

Know the derivative of tan x to remember the integral of sec² x.
Review trig identities before integrating complex expressions.
Watch for linear arguments requiring division by k.

通过记住 tan x 的导数来记住 sec² x 的积分。
积分复杂表达式前先复习三角恒等式。
注意线性参数需要除以 k。

6. Inverse Trig: 1/(1+x²) and 1/√(1-x²) | 反三角函数：1/(1+x²) 与 1/√(1-x²)

Inverse trigonometric functions appear in integrals that are not obviously trigonometric. The integral of 1/(1+x²) is arctan x + C, and the integral of 1/√(1-x²) is arcsin x + C.

反三角函数出现在一些看起来不明显的三角积分中。1/(1+x²) 的积分是 arctan x + C，1/√(1-x²) 的积分是 arcsin x + C。

∫ 1/(1+x²) dx = arctan x + C; ∫ 1/√(1-x²) dx = arcsin x + C

These formulas are used when integrating rational functions that can be rearranged into these standard forms, often after completing the square.

这些公式用于化简为上述标准形式的有理函数积分，通常需要先配方。

For example, ∫ 1/(4+x²) dx can be rewritten as (1/2) arctan(x/2) + C.

例如，∫ 1/(4+x²) dx 可以改写为 (1/2) arctan(x/2) + C。

Use completing the square to match the standard form.
Remember the scaling factor when the coefficient differs from 1.
Check the domain for inverse trig functions.

使用配方法将积分化为标准形式。
当系数不是 1 时，记得乘以缩放因子。
注意反三角函数的定义域。

7. Integration Table of Common Functions | 常见标准函数积分表

The following table consolidates the most important standard integrals for quick revision. You should practise until these are memorised.

下表汇总了最重要的标准积分公式，便于快速复习。你应该反复练习，直到熟练掌握。

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

Use this table as a checklist. In exams, you are expected to know these results without a formula booklet.

使用这张表作为检查清单。在考试中，你应当无需公式册就能记住这些结果。

8. Worked Examples | 例题精讲

Example 1: Find ∫ (3x² – 4x + 5) dx.

例题 1：求 ∫ (3x² – 4x + 5) dx。

Apply the power rule term by term: ∫ 3x² dx = x³, ∫ -4x dx = -2x², and ∫ 5 dx = 5x. Therefore the answer is x³ – 2x² + 5x + C.

逐项应用幂函数积分公式：∫ 3x² dx = x³，∫ -4x dx = -2x²，∫ 5 dx = 5x。所以结果为 x³ – 2x² + 5x + C。

Example 2: Evaluate ∫₀¹ (e²ˣ + 1/x) dx.

例题 2：计算 ∫₀¹ (e²ˣ + 1/x) dx。

Integrate e²ˣ to get e²ˣ/2 and integrate 1/x to get ln|x|. Then substitute the limits: (e²/2 + ln 1) – (e⁰/2 + ln 1) = e²/2 – 1/2 = (e² – 1)/2.

e²ˣ 的积分为 e²ˣ/2，1/x 的积分为 ln|x|。代入上下限：(e²/2 + ln 1) – (e⁰/2 + ln 1) = e²/2 – 1/2 = (e² – 1)/2。

Example 3: Find the antiderivative of sin(3x) + sec²(2x).

例题 3：求 sin(3x) + sec²(2x) 的原函数。

The integral is -cos(3x)/3 + tan(2x)/2 + C. Use the linear argument rule for each term.

积分为 -cos(3x)/3 + tan(2x)/2 + C。各项分别使用线性参数规则即可。

9. Common Traps and How to Avoid Them | 常见陷阱与应对策略

One common mistake is applying the power rule to

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Common Standard Function Integrals | 常见标准函数的积分公式