📚 Exercise 13A.2: Basic Integration Techniques | 基础积分技巧练习13A.2
This section revisits the core ideas behind antidifferentiation as covered in Exercise 13A.2 of the IB Mathematics syllabus. You will learn to reverse differentiation rules for polynomials, trigonometric functions, and exponentials, while paying close attention to the constant of integration. A solid grasp of these fundamentals is essential for tackling definite integrals, area problems, and further calculus topics later in the course.
本节回顾 IB 数学课程中练习13A.2所涵盖的反导数核心思想。你将学习如何逆向运用多项式、三角函数和指数函数的微分法则，并特别关注积分常数。牢固掌握这些基础对于后续处理定积分、面积问题以及更深入的微积分主题至关重要。
1. What Is an Antiderivative? | 什么是反导数？
A function F(x) is called an antiderivative of f(x) if F ' ivative of x³ is 3x², an antiderivative of 3x² is x³. However, any function of the form x³ + C, where C is an arbitrary constant, is also an antiderivative because the derivative of a constant is zero.
如果 F '(x) = f(x)，则称函数 F(x) 是 f(x) 的一个反导数。例如，由于 x³ 的导数是 3x²，因此 3x² 的一个反导数就是 x³。然而，任何形如 x³ + C（C 为任意常数）的函数也都是反导数，因为常数的导数为零。
otation is central to Exercise 13A.2, where you practise writing the general antiderivative for a variety of basic functions.
找出所有反导数的过程称为不定积分，用积分号 ∫ 表示。因此我们记作 ∫ f(x) dx = F(x) + C。这一记号是练习13A.2的核心，练习中你会写出各类基本函数的一般反导数。
∫ f(x) dx = F(x) + C, where F '(x) = f(x)
2. The Power Rule for Integration | 幂函数的积分法则
The power rule for integration is the reverse of the power rule for differentiation. For any real number n ≠ –1, integrating xⁿ adds one to the exponent and divides by the new exponent.
幂函数的积分法则是幂函数微分法则的逆运算。对于任何实数 n ≠ –1，对 xⁿ 积分时将指数加1，再除以新的指数。
∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, n ≠ –1
ge family of polynomials and algebraic expressions. For example, ∫ x² dx = x³/3 + C, and ∫ √x /3)x³/² + C. Notice that the constant C must always be included in indefinite integration.
这条规则涵盖了大量的多项式与代数表达式。例如，∫ x² dx = x³/3 + C，以及 ∫ √x dx = ∫ x½ dx = x³/² / (3/2) + C = (2/3)x³/² + C。注意不定积分中必须始终包含常数 C。
When n = – e power rule gives division by zero, so we need a special result: ∫ x⁻¹ dx = ∫ 1/x dx = ln|x| + C. The absolute value ensures the expression is defined for negative x as well.
当 n = –1 时，幂法则会导致除以零，因此需要一个特殊结果：∫ x⁻¹ dx = ∫ 1/x dx = ln|x| + C。绝对值保证了该表达式对负 x 也有定义。
3. Integrating Trigonometric Functions | 三角函数的积分
/dx(sin x) = cos x, we immediately obtain ∫ cos x dx = sin x + C. Similarly, d/dx(cos x) = –sin x gives ∫ sin x dx = –cos x + C.
基本三角函数的积分直接由正弦和余弦的导数得出。因为 d/dx(sin x) = cos x，立刻得到 ∫ cos x dx = sin x + C。类似地，d/dx(cos x) = –sin x 给出 ∫ sin x dx = –cos x + C。
In Exercise 13A.2 you will often encounter linear arguments such as sin(ax) or cos(ax). By mentally differentiating –cos(ax)/a you can see that the derivative is sin(ax), which gives the integral formula.
在练习13A.2中，你经常会遇到诸如 sin(ax) 或 cos(ax) 这样的线性变量。通过心算对 –cos(ax)/a 求导，可以发现其导数为 sin(ax)，从而得到积分公式。
∫ sin(ax) dx = –cos(ax)/a + C
∫ cos(ax) dx = sin(ax)/a + C
It is worth memorising that the integral of sin is negative cosine, while the integral of cosine is positive sine. The factor 1/a appears naturally when applying the chain rule in reverse.
值得记住的是，sin 的积分是负的 cosine，而 cosine 的积分是正的 sine。逆向应用链式法则时，自然会出现因子 1/a。
4. Integrating Exponential Functions | 指数函数的积分
The exponential function eˣ is unique because its derivative is itself, so its integral is also itself plus the constant of integration.
指数函数 eˣ 的特殊之处在于其导数就是它本身，因此它的积分也是它本身加上积分常数。
∫ eˣ dx = eˣ + C
For a more general exponential eᵏˣ, where k is a constant, the chain rule gives a derivative of k eᵏˣ. Reversing this, we divide by k when integrating.
对于更一般的指数 eᵏˣ（k 为常数），链式法则给出的导数为 k eᵏˣ。逆向操作，积分时需要除以 k。
∫ eᵏˣ dx = (1/k) eᵏˣ + C
This rule is frequently tested in basic integration exercises. For instance, ∫ e³ˣ dx = (1/3) e³ˣ + C. Always check your answer by differentiating – you should recover the original integrand.
这条规则在基础积分练习中经常出现。例如，∫ e³ˣ dx = (1/3) e³ˣ + C。始终通过求导来检验你的答案——你应该得到原来的被积函数。
5. The Constant of Integration | 积分常数
The "+ C" is not a trivial detail; it represents an entire family of functions that differ by a constant. When you are given an initial condition or a point on the curve, you can determine the specific value of C. Without this constant, the antiderivative is incomplete as a general solution.
“+ C”不是一个可有可无的细节；它表示一簇相差一个常数的函数。当给定初始条件或曲线上一点时，你可以确定 C 的具体值。如果没有这个常数，反导数作为通解就是不完整的。
In the context of Exercise 13A.2, you are usually asked to find the indefinite integral without additional conditions, so "+ C" must always be written. Forgetting the constant is one of the most common mistakes at this stage.
在练习13A.2的情境中，通常要求你在没有附加条件的情况下求不定积分，因此必须始终写上“+ C”。忘记这个常数是这个阶段最常见的错误之一。
6. Simplifying Before Integrating | 积分前的化简
Many integrands in Exercise 13A.2 look complicated at first glance but become straightforward after algebraic simplification. Expand products, separate fractions, or rewrite radicals as fractional exponents before applying the integration rules.
练习13A.2中的许多被积函数乍看之下很复杂，但经过代数化简后就会变得简单。在应用积分法则之前，先展开乘积、拆分分式，或将根式改写为分数指数形式。
For example, ∫ (x²+1)/x dx can be written as ∫ (x + 1/x) dx = x²/2 + ln|x| + C. Similarly, ∫ (√x – 2)² dx becomes ∫ (x – 4√x + 4) dx after expanding, which can then be integrated term by term.
例如，∫ (x²+1)/x dx 可以写成 ∫ (x + 1/x) dx = x²/2 + ln|x| + C。类似地，∫ (√x – 2)² dx 展开后变为 ∫ (x – 4√x + 4) dx，然后可以逐项积分。
Developing the habit of rewriting the integrand in a form that matches standard rules will greatly increase your accuracy and speed.
养成将被积函数改写为符合标准规则形式的习惯，将极大提高你的准确性和速度。
7. Recognising Patterns and Reverse Chain Rule | 识别模式与逆向链式法则
Exercise 13A.2 begins to build your intuition for the reverse chain rule. If an integrand looks like the derivative of a known composite function, you can guess the antiderivative and adjust constants.
练习13A.2开始培养你对逆向链式法则的直觉。如果被积函数看起来像某个已知复合函数的导数，你可以猜出反导数并调整常数。
A simple case is ∫ 2x eˣ² dx. Since the derivative of eˣ² is 2x eˣ², the integral is eˣ² + C. More generally, ∫ f '(x) eᶠ⁽ˣ⁾ dx = eᶠ⁽ˣ⁾ + C. Similar patterns exist for trigonometric functions: ∫ f '(x) cos(f(x)) dx = sin(f(x)) + C.
一个简单的例子是 ∫ 2x eˣ² dx。因为 eˣ² 的导数是 2x eˣ²，所以积分就是 eˣ² + C。更一般地，∫ f '(x) eᶠ⁽ˣ⁾ dx = eᶠ⁽ˣ⁾ + C。对于三角函数也有类似的模式：∫ f '(x) cos(f(x)) dx = sin(f(x)) + C。
While Exercise 13A.2 may not involve heavy substitution yet, being able to spot these patterns prepares you for integration by substitution in later exercises.
虽然练习13A.2可能还不涉及复杂的代换，但能够识别这些模式会为你后续的换元积分法做好准备。
8. Common Mistakes and How to Avoid Them | 常见错误及如何避免
power rule to 1/x and writing x⁰/0, which is meaningless. Always recall that ∫ 1/x dx = ln|x| + C. Another frequent slip is mishandling the coefficient when integrating eᵏˣ or sin(ax) – forgetting to divide by k or a.
一个典型错误是将幂法则应用于 1/x 并写出 x⁰/0，这是无意义的。务必牢记 ∫ 1/x dx = ln|x| + C。另一个常见失误是在积分 eᵏˣ 或 sin(ax) 时处理系数不当——忘记除以 k 或 a。
Students also sometimes omit the constant C or write incorrect signs for trigonometric integrals. A quick mental differentiation check can catch nearly all these errors. Additionally, when working with fractional exponents, be careful with arithmetic – √x is x½, and the new exponent becomes 3/2, leading to the factor 2/3.
学生有时还会漏掉常数 C，或者写错三角函数的积分符号。一个快速的求导心算检查几乎能发现所有这类错误。此外，处理分数指数时要小心运算——√x 是 x½，新的指数变为 3/2，从而得到因子 2/3。
Finally, remember that integration is not distributive over multiplication. You cannot integrate a product by integrating each factor separately; you must first rewrite the expression into a sum of simpler terms if possible.
最后，记住积分对乘法不满足分配律。不能通过分别积分每个因子来积分一个乘积；你必须尽可能先将表达式改写为较简单项的和。
9. Practice Examples in the Style of Exercise 13A.2 | 练习13A.2风格的实例
Working through varied examples builds fluency. Below are representative problems of the type found in Exercise 13A.2. Try each one, then check by differentiation.
通过不同类型的例子进行练习可以培养熟练度。以下是与练习13A.2题型类似的代表性题目。尝试求解每一道题，然后通过求导加以检查。
| Integral | Answer |
|---|---|
| ∫ (4x³ – 6x + 5) dx | x⁴ – 3x² + 5x + C |
| ∫ (2 cos x – 3 sin x) dx | 2 sin x + 3 cos x + C |
| ∫ (e²ˣ + 1/x) dx | (1/2)e²ˣ + ln|x| + C |
| ∫ (√t + 1/√t) dt | (2/3)t³/² + 2√t + C |
| ∫ (3 sin(2x) – cos(4x)) dx | –(3/2) cos(2x) – (1/4) sin(4x) + C |
Notice how each answer includes the constant C and the necessary coefficients. The more you practise, the more natural these patterns will become.
注意每个答案都包含了常数 C 和必要的系数。你练习得越多，这些模式就会变得越自然。
10. Linking Indefinite Integrals to Definite Integrals | 将不定积分与定积分联系起来
valuate a definite integral ∫ₐᵇ f(x) dx, we find an antiderivative F(x), then compute F(b) – F(a).
尽管练习13A.2侧重于不定积分，但你所掌握的技巧构成了定积分的基础。微积分基本定理告诉我们，要计算定积分 ∫ₐᵇ f(x) dx，我们须先找出一个反导数 F(x)，然后计算 F(b) – F(a)。
Without a fluent command of the antiderivatives covered in this exercise, moving on to area calculations, kinematics problems, and other applications would be extremely difficult. Treat every integration problem now as an opportunity to strengthen that essential bridge.
如果对本练习所涵盖的反导数没有熟练的掌握，后续进行面积计算、运动学问题及其他应用将极其困难。现在就把每一道积分题目当作巩固这一重要桥梁的机会。
In the next sections of the textbook, you will encounter integration by substitution and integration by parts, both of which rely heavily on recognising basic antiderivative forms instantly. Building a solid repertoire now will pay dividends later.
在教材的后续章节中，你将会碰到换元积分法和分部积分法，两者都严重依赖于即时识别基本反导数形式的能力。现在建立一个扎实的积分库，日后将大有裨益。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图