Mastering the Table of Standard Antiderivatives: Construction and Applications | 常用原函数表：构建方法与应用

📚 Mastering the Table of Standard Antiderivatives: Construction and Applications | 常用原函数表：构建方法与应用

The concept of the antiderivative, also known as the indefinite integral, is one of the cornerstones of calculus. In the IB Mathematics: Analysis and Approaches course, students are expected to integrate a wide range of functions quickly and accurately. Rather than memorising the rule table blindly, a deeper understanding of how the table is constructed from known derivatives empowers you to reconstruct formulas under exam pressure, verify your answers by differentiation, and tackle unfamiliar integrals with confidence.

原函数（又称不定积分）的概念是微积分的基石之一。在 IB 数学：分析与方法（AA）课程中，学生需要快速且准确地积分多种函数。与其盲目记忆公式表，不如深入理解这些公式是如何从已知导数构建而来的。这不仅能帮助你在考试压力下重建公式，也能让你通过求导来检验答案，并自信地处理不熟悉的不定积分。

1. From Derivatives to Antiderivatives | 从导数到原函数

Every formula in the table of standard antiderivatives is obtained by reversing a known derivative rule. For example, since the derivative of x² is 2x, we know that an antiderivative of 2x is x², so ∫2x dx = x² + C. The additive constant C is essential because differentiating any constant gives zero.

原函数表中的每一个公式都是通过反向使用已知的求导规则得到的。例如，因为 x² 的导数是 2x，所以 2x 的一个原函数是 x²，因此 ∫2x dx = x² + C。这里的积分常数 C 必不可少，因为任意常数求导结果都是零。

The most basic construction rule is the power rule for integration. For any real number n ≠ −1, we have ∫xⁿ dx = xⁿ⁺¹ / (n+1) + C. You can verify this by differentiating the right-hand side: the derivative is xⁿ, exactly the integrand.

最基本的构建规则是幂函数积分法则。对任何实数 n ≠ −1，有 ∫xⁿ dx = xⁿ⁺¹ / (n+1) + C。你可以在右侧求导验证：其导数恰好就是被积函数 xⁿ。

This rule also handles negative and fractional exponents. For instance, ∫x⁻² dx = x⁻¹ / (−1) + C = −x⁻¹ + C, and ∫√x dx = ∫x^(1/2) dx = (2/3)x^(3/2) + C. In general, the table is powerful because it covers all rational powers in one compact equation.

这个法则同样适用于负指数和分数指数。例如，∫x⁻² dx = x⁻¹ / (−1) + C = −x⁻¹ + C；∫√x dx = ∫x^(1/2) dx = (2/3)x^(3/2) + C。一般来说，这张表之所以强大，正是因为它用一条紧凑的方程覆盖了所有有理数幂。

2. Exponential and Logarithmic Functions | 指数函数与对数函数

Exponential functions are their own derivatives in the base-e case. Since d/dx(eˣ)=eˣ, we immediately get ∫eˣ dx = eˣ + C. More generally, for a constant base a > 0, a ≠ 1, we have d/dx(aˣ)=aˣ ln a, so ∫aˣ dx = aˣ / ln a + C.

指数函数在底数为 e 时是其自身的导数。因为 d/dx(eˣ)=eˣ，我们立即得到 ∫eˣ dx = eˣ + C。更一般地，对常数底 a > 0 且 a ≠ 1，有 d/dx(aˣ)=aˣ ln a，因此 ∫aˣ dx = aˣ / ln a + C。

The natural logarithm completes the power rule at n = −1. Since d/dx(ln|x|)=1/x for x≠0, we have ∫x⁻¹ dx = ∫1/x dx = ln|x| + C. This single exception is heavily tested in exams, especially when integrating rational functions.

自然对数为 n = −1 时的幂规则补上了缺口。因为 d/dx(ln|x|)=1/x（x≠0），所以 ∫x⁻¹ dx = ∫1/x dx = ln|x| + C。这个唯一例外在考试中反复出现，特别是在积分有理函数时。

In IB questions, you may need to combine these rules. For example, ∫(2ˣ + 3/x) dx = 2ˣ / ln2 + 3 ln|x| + C. Always keep the absolute value inside the logarithm when the integrand could be negative.

在 IB 题目中，你可能需要结合这些规则。例如，∫(2ˣ + 3/x) dx = 2ˣ / ln2 + 3 ln|x| + C。当被积函数可能为负时，记得在对数内保留绝对值符号。

3. Trigonometric Functions | 三角函数

The standard trigonometric antiderivatives come directly from derivative rules. For instance, d/dx(sin x)=cos x, so ∫cos x dx = sin x + C; and d/dx(cos x)=−sin x, so ∫sin x dx = −cos x + C.

标准三角函数的原函数直接来自求导规则。例如，d/dx(sin x)=cos x，所以 ∫cos x dx = sin x + C；d/dx(cos x)=−sin x，所以 ∫sin x dx = −cos x + C。

For tangent and secant, the constructions are slightly less direct. Use d/dx(ln|sec x|)=tan x to obtain ∫tan x dx = ln|sec x| + C, and d/dx(ln|sec x + tan x|)=sec x to obtain ∫sec x dx = ln|sec x + tan x| + C. Other reciprocal trigonometric functions similarly involve logarithms.

对于正切和正割，构造过程稍微间接一些。利用 d/dx(ln|sec x|)=tan x 可得 ∫tan x dx = ln|sec x| + C；利用 d/dx(ln|sec x + tan x|)=sec x 可得 ∫sec x dx = ln|sec x + tan x| + C。其余互为倒数的三角函数也类似地涉及对数。

Do not forget the standard constant multiples: ∫csc² x dx = −cot x + C and ∫sec² x dx = tan x + C. These two appear often in Paper 2, often disguised inside more complex products.

别忘了标准常数倍情形：∫csc² x dx = −cot x + C，以及 ∫sec² x dx = tan x + C。这两条在第二卷中经常出现，有时会隐藏在某些更复杂的乘积表达式中。

4. Inverse Trigonometric Functions | 反三角函数

The derivatives of inverse trigonometric functions give rise to two classic antiderivatives: ∫1/√(1−x²) dx = arcsin x + C and ∫1/(1+x²) dx = arctan x + C. These are often restated in the transformed forms ∫1/√(a²−x²) dx = arcsin(x/a) + C and ∫1/(a²+x²) dx = (1/a) arctan(x/a) + C.

反三角函数的导数产生两个经典原函数：∫1/√(1−x²) dx = arcsin x + C 和 ∫1/(1+x²) dx = arctan x + C。它们通常以变形形式出现：∫1/√(a²−x²) dx = arcsin(x/a) + C 以及 ∫1/(a²+x²) dx = (1/a) arctan(x/a) + C。

These formulas are vital for integrating functions of the form 1/(quadratic) or 1/√(quadratic). On IB exams, they often appear after a linear

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