📚 E Tables of Indefinite and Definite Integrals | 不定积分与定积分表
Mastering integration is essential for IB Mathematics, and a well‑organised table of integrals saves time during revision and exams. This article presents a comprehensive set of indefinite and definite integral formulas aligned with the IB Analysis & Approaches and Applications & Interpretation syllabi. Use these tables to check standard results, recognise patterns, and solve problems efficiently.
掌握积分对于IB数学至关重要，而一个组织良好的积分表可以在复习和考试中节省时间。本文提供了一套与IB数学分析与方法和应用与解释课程大纲相符的、全面的不定积分和定积分公式表。利用这些表格可以核对标准结果、识别模式并高效解题。
1. Introduction to Integral Tables | 积分表简介
Integral tables list antiderivatives of common functions. They are useful when direct integration is not obvious, or when verifying answers. For indefinite integrals, always add the constant of integration C. For definite integrals, the limits of integration turn an antiderivative into a numerical value.
积分表列出了常见函数的原函数。当直接积分不明显或需要验证答案时，它们非常有用。对于不定积分，始终要加上积分常数 C。对于定积分，积分上下限将原函数转化为一个数值。
In the tables below, all variables represent real numbers unless stated otherwise, and the natural logarithm is denoted by ln. The integration constant C is omitted in the table for brevity, but it must be included in final answers.
在下表中，除非另有说明，所有变量均表示实数，自然对数用 ln 表示。为简洁起见，表中省略了积分常数 C，但在最终答案中必须包含。
2. Basic Indefinite Integrals | 基本不定积分表
These formulas form the foundation for all integration techniques. Memorising them is strongly recommended. All trigonometric arguments are in radians.
这些公式是所有积分技巧的基础，强烈建议记背。所有三角函数参数均以弧度为单位。
| Function | Indefinite Integral |
|---|---|
| k (constant) | ∫ k dx = kx + C |
| xn (n ≠ −1) | ∫ xn dx = xn+1/(n+1) + C |
| 1/x | ∫ 1/x dx = ln|x| + C |
| ex | ∫ ex dx = ex + C |
| ax (a>0, a≠1) | ∫ ax dx = ax/ln a + C |
| sin x | ∫ sin x dx = −cos x + C |
| cos x | ∫ cos x dx = sin x + C |
| sec2 x | ∫ sec2 x dx = tan x + C |
| csc2 x | ∫ csc2 x dx = −cot x + C |
| sec x tan x | ∫ sec x tan x dx = sec x + C |
| csc x cot x | ∫ csc x cot x dx = −csc x + C |
| 1/√(1−x2) | ∫ 1/√(1−x2) dx = arcsin x + C |
| 1/(1+x2) | ∫ 1/(1+x2) dx = arctan x + C |
Although the constant C is omitted in the table, you must add it in your answers for indefinite integrals. For definite integrals, the constant cancels out.
尽管表中省略了常数 C，但你必须在不定积分的答案中添加。对于定积分，常数会被抵消。
3. Trigonometric Integrals | 三角函数积分
Beyond the basic trigonometric integrals, some functions require the use of identities or result in logarithmic forms. These integrals appear frequently in HL problems.
除了基本三角函数积分外，某些函数需要利用恒等式或产生对数形式。这些积分在HL问题中经常出现。
| Function | Indefinite Integral |
|---|---|
| tan x | ∫ tan x dx = ln|sec x| + C = −ln|cos x| + C |
| cot x | ∫ cot x dx = ln|sin x| + C |
| sec x | ∫ sec x dx = ln|sec x + tan x| + C |
| csc x | ∫ csc x dx = −ln|csc x + cot x| + C = ln|tan(x/2)| + C |
| sin2 x | ∫ sin2 x dx = x/2 − (sin 2x)/4 + C |
| cos2 x | ∫ cos2 x dx = x/2 + (sin 2x)/4 + C |
When integrating even powers of sine or cosine, use the power‑reduction identities to rewrite the integrand before integrating term by term.
当对正弦或余弦的偶次幂积分时，应先用降幂恒等式改写被积函数，再逐项积分。
4. Inverse Trigonometric Integrals | 反三角函数积分
These generalisations include a constant coefficient inside the square root or denominator. They are especially useful for integrals arising from trigonometric substitution.
这些推广形式在平方根或分母中包含了常数系数。它们对于三角代换产生的积分尤为有用。
| Form | Integral |
|---|---|
| 1/√(a2−x2) | ∫ dx/&radic；(a2−x2) = arcsin(x/a) + C |
| 1/(a2+x2) | ∫ dx/(a2+x2) = (1/a) arctan(x/a) + C |
| 1/(x√(x2−a2))
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