📚 Exercise 22C.2: Mastering Integration by Substitution | 练习22C.2：掌握代换积分法
Exercise 22C.2 in the IB Mathematics curriculum is a dedicated practice set designed to build fluency in one of the most powerful techniques of integration: integration by substitution. This method, often called u‑substitution, is the reverse of the chain rule for differentiation and is essential for evaluating a wide range of integrals that appear in both Standard Level and Higher Level assessments. In this article we unpack the key ideas behind the exercise, explain how to think when choosing a substitution, walk through representative examples mirroring those you will meet, and highlight common mistakes to avoid. Our goal is to help you turn Exercise 22C.2 from a routine drill into real mastery that will serve you well in Paper 1 and Paper 2 questions.
练习22C.2是IB数学课程中专门为强化一种最强大的积分技巧——代换积分法而设计的训练集。这种方法常被称为u代换，是微分链式法则的逆运算，对于解答标准水平和高水平考试中出现的各种积分至关重要。本文将剖析该练习背后的核心思想，解释选择代换时的思考方式，逐步演示与你将要遇到的题目相似的代表性例子，并指出需要避免的常见错误。我们的目标是将练习22C.2从普通的重复训练转变为真正的精通，帮助你在试卷一和试卷二中取得好成绩。
1. Understanding Integration by Substitution | 理解代换积分法
Integration by substitution is essentially a method for simplifying an integral by introducing a new variable, usually denoted u, to reduce a complicated integrand into a standard form that we can recognise and integrate. The rule is formally derived from the chain rule for differentiation: if we set u = g(x), then du = g'(x) dx, which allows us to rewrite ∫ f(g(x)) g'(x) dx as ∫ f(u) du. The success of the method hinges on spotting a function and its derivative (or a constant multiple thereof) inside the integral.
代换积分法本质上是通过引入一个新变量（通常用u表示）来简化被积函数，将复杂的式子化成我们可以识别并积分的标准形式。其规则严格来源于微分的链式法则：如果设 u = g(x)，则 du = g'(x) dx，这就使我们能够将 ∫ f(g(x)) g'(x) dx 重写为 ∫ f(u) du。该方法成功的关键在于能否在积分中认出某个函数及其导数（或导数的常数倍）。
2. The Reverse Chain Rule Perspective | 反向链式法则视角
Think of substitution as the counterpart to differentiating a composite function. When we differentiate sin(x²), we obtain 2x cos(x²). If we are asked to integrate ∫ 2x cos(x²) dx, we notice the derivative of the inner function x² is exactly 2x, which sits conveniently next to cos(x²). By letting u = x², the integral becomes ∫ cos(u) du = sin(u) + C = sin(x²) + C. The exercise 22C.2 is full of such set-ups, purposely crafted to train your eye to see this reverse chain rule pattern instantly.
你可以将代换法视为复合函数微分的逆运算。当我们对 sin(x²) 求导时，得到 2x cos(x²)。如果要求计算 ∫ 2x cos(x²) dx，我们会发现内层函数 x² 的导数正好是 2x，恰恰位于 cos(x²) 旁边。令 u = x²，积分就变成了 ∫ cos(u) du = sin(u) + C = sin(x²) + C。练习22C.2中充满了这类设计，目的就是训练你的眼睛能立刻认出这种反向链式法则的模式。
3. Choosing the Right Substitution u | 选择正确的代换变量u
There is no mechanical recipe that works for every integral, but a good starting point is to look for an “inner function” whose derivative also appears in the integrand. For polynomial powers like ∫ x (3x² + 5)⁴ dx, the natural choice is u = 3x² + 5, because du = 6x dx and we already have an x dx present. Frequently the substitution can be a linear expression raised to a power, a denominator, an exponent, or the argument of a trigonometric or logarithmic function. In Exercise 22C.2 you will practise making these decisions and adjusting constants to match du.
没有一种机械性的方法适用于所有积分，但一个良好的出发点是寻找一个“内层函数”，其导数也出现在被积函数中。对于 ∫ x (3x² + 5)⁴ dx 这类多项式幂，自然的选择是 u = 3x² + 5，因为 du = 6x dx，而我们已经有了一个 x dx。很多时候代换对象可以是带幂的线性式、分母、指数，或是三角函数或对数函数的自变量。在练习22C.2中，你将练习做出这些选择，并调整常数以便与 du 匹配。
4. Step‑by‑Step Example 1: Polynomial Power | 步骤示例1：多项式幂
Consider an integral similar to those in the exercise: ∫ 4x (2x² − 3)⁵ dx. We identify the inner function g(x) = 2x² − 3, so set u = 2x² − 3. Then du/dx = 4x, which gives du = 4x dx. Notice that 4x dx appears exactly in the integrand. The integral transforms into ∫ u⁵ du, which yields (1/6) u⁶ + C = (1/6)(2x² − 3)⁶ + C. The key steps are: define u, find du, replace all x‑terms, integrate in u, and back‑substitute.
考虑一个类似练习中的积分：∫ 4x (2x² − 3)⁵ dx。我们先找出内层函数 g(x) = 2x² − 3，于是设 u = 2x² − 3。那么 du/dx = 4x，得到 du = 4x dx。注意 4x dx 正好出现在被积函数中。积分就变成 ∫ u⁵ du，结果为 (1/6) u⁶ + C = (1/6)(2x² − 3)⁶ + C。关键步骤是：定义 u；求 du；替换所有含 x 的项；对 u 积分；再代回原变量。
5. Worked Example 2: Integrating Trigonometric Functions | 示例2：三角函数的积分
A typical IB‑style question is ∫ sin³x cos x dx. Here the derivative of sin x is cos x, which sits alongside the power of sine. Let u = sin x, then du = cos x dx. The integral becomes ∫ u³ du = (1/4) u⁴ + C = (1/4) sin⁴x + C. Exercise 22C.2 explicitly helps you recognise this pattern with various powers of sine and cosine. Always check whether the integrand contains a function and its exact derivative; if the derivative is off by a constant factor you can still proceed by multiplying or dividing by that constant.
一个典型的IB题目是 ∫ sin³x cos x dx。这里 sin x 的导数是 cos x，它紧挨着正弦的幂次。令 u = sin x，则 du = cos x dx。积分变成 ∫ u³ du = (1/4) u⁴ + C = (1/4) sin⁴x + C。练习22C.2明确地帮助你识别这种包含正弦余弦各种幂次的模式。要始终检查被积函数是否包含一个函数及其精确导数；如果导数相差一个常数倍，仍然可以通过乘以或除以该常数来继续计算。
6. Worked Example 3: Exponential Integrals | 示例3：指数积分
Exponential integrals such as ∫ x e^(x²) dx are extremely common. The derivative of the exponent x² is 2x, so we set u = x², du = 2x dx. The integrand has x dx, which is half of du, so we rewrite: (1/2) ∫ e^u du = (1/2) e^u + C = (1/2) e^(x²) + C. Other exponential patterns include ∫ e^(ax+b) dx, which is a straightforward linear substitution, and more involved cases where the exponent is a trigonometric function, which also appear in the practice set.
形如 ∫ x e^(x²) dx 的指数积分非常常见。指数 x² 的导数是 2x，因此设 u = x²，du = 2x dx。被积函数中有 x dx，是 du 的一半，于是改写为 (1/2) ∫ e^u du = (1/2) e^u + C = (1/2) e^(x²) + C。其他指数模式还包括 ∫ e^(ax+b) dx，这是一个简单的线性代换，以及指数部分为三角函数的更复杂情形，这些也都会出现在练习集中。
7. Definite Integrals and Changing Limits | 定积分与变换积分限
When applying substitution to a definite integral ∫ₐᵇ f(x) dx, you have two options: either integrate in u and then substitute back to x before evaluating at the original limits, or—more efficiently—change the limits when you change the variable. If u = g(x), the new limits become uₗ = g(a) and uᵤ = g(b). Then evaluate ∫_{uₗ}^{uᵤ} f(u) du directly. Exercise 22C.2 includes definite integrals to ensure you are comfortable with both methods. Carelessness with limit conversion is a major source of errors, so always write the new limits clearly.
对定积分 ∫ₐᵇ f(x) dx 使用代换法时，你有两种选择：一是在 u 下积出结果后再代回 x，然后用原来的上下限求值；二是——更高效的办法——在换变量的同时变换上下限。若 u = g(x)，新的积分限就变为 uₗ = g(a) 和 uᵤ = g(b)。然后直接计算 ∫_{uₗ}^{uᵤ} f(u) du。练习22C.2包含了定积分，以确保你对这两种方法都感到得心应手。积分限转换时的疏忽是主要的错误来源，因此要始终清晰地写出新的上下限。
8. Adjusting for Constants and Missing Factors | 常数与缺失因子的调整
Often the derivative of your chosen u is not exactly present; instead a constant multiple appears. For ∫ x √(x²+1) dx, let u = x²+1, du = 2x dx. We have x dx, which is (1/2) du. Thus the integral becomes (1/2) ∫ √u du. The factor 1/2 is simply moved outside. Never forget to compensate for the missing constant—it is one of the most frequent slip‑ups in timed exam settings. The exercises deliberately mix cases with and without exact derivative matches to sharpen this skill.
很多时候，所选 u 的导数并非原封不动地出现，而是相差一个常数倍。对 ∫ x √(x²+1) dx，令 u = x²+1，du = 2x dx。我们得到的是 x dx，即 (1/2) du。于是积分变为 (1/2) ∫ √u du。因子 1/2 可以简单地提到外面。永远不要忘记补偿缺失的常数——这是限时考试中最常见的失误之一。练习中刻意混杂了导数完全匹配与不完全匹配的情形，以打磨这一技能。
9. Substitution with Rational and Radical Functions | 有理式和根式函数的代换
Substitution is particularly handy for integrals involving square roots and rational functions where the denominator’s derivative is visible. For ∫ (2x+1)/(x²+x+3) dx, set u = x²+x+3, then du = (2x+1) dx, exactly the numerator. The integral reduces to ∫ (1/u) du = ln|u| + C. Similarly, for ∫ 1/√(1−x²) dx the standard trigonometric substitution is taught later, but simpler radical functions like ∫ x √(4−x²) dx are mastered in this exercise using u = 4−x². Recognising these forms opens the door to more complex integrations.
代换法对于涉及平方根和分母的导数可以被看见的有理式积分尤其便捷。对于 ∫ (2x+1)/(x²+x+3) dx，设 u = x²+x+3，则 du = (2x+1) dx，正好是分子。积分化简为 ∫ (1/u) du = ln|u| + C。类似地，虽然 ∫ 1/√(1−x²) dx 的标准三角函数代换会在后面学到，但在本练习中你将通过 u = 4−x² 掌握像 ∫ x √(4−x²) dx 这样较简单的根式函数。认识这些形式可为更复杂的积分打开大门。
10. Common Errors and How to Avoid Them | 常见错误与避免方法
Watch out for the following pitfalls: forgetting to replace dx with du/(g'(x)) entirely, leaving a mixture of x and u in the same integrand; mishandling constants when du is not a perfect match; dropping the absolute value in logarithmic integrals; and, for definite integrals, neglecting to update the limits. In Exercise 22C.2, many problems are structured to test exactly these traps. After solving, quickly differentiate your result mentally to check if you recover the original integrand—this self‑check can catch most slip‑ups and build lasting confidence.
要当心以下陷阱：忘记将 dx 完全替换为 du/(g'(x))，导致同一被积函数中同时出现 x 和 u；当 du 不是完美匹配时错误处理常数；在对数积分中漏掉绝对值；以及在定积分中忘记更换积分限。练习22C.2中的许多题目正是为了检验这些陷阱而设计的。解完题后，在脑中快速求导，检查是否能回到原被积函数——这一自我检验能发现大部分失误，并建立持久的自信。
11. Linking Substitution to IB Exam Questions | 代换积分与IB考试题目的连结
IB examination papers frequently embed substitution within longer problems—finding areas under curves, solving differential equations, or computing volumes of revolution. A typical Paper 2 question might ask you to integrate a function obtained from a previous differentiation or to set up an integral from a graph and then evaluate it using an appropriate substitution. The fluency you gain from completing Exercise 22C.2 means you can recognise the necessary substitution quickly and allocate more time to the application part of the problem, which is often where higher marks are awarded.
IB考试中经常将代换法嵌入更长的题目中——求曲线下的面积、解微分方程或计算旋转体体积。典型的试卷二问题可能会要求你积分一个从前一步微分得到的函数，或根据图像建立积分再用适当的代换求值。完成练习22C.2后所获得的熟练度，意味着你能迅速识别必要的代换，从而将更多时间分配给问题的应用部分，而这部分往往是获得高分的关键。
12. Practice Strategy for Exercise 22C.2 | 练习22C.2的训练策略
Start by working through the first few problems with your notes open, deliberately writing each step: declare u, compute du, adjust constants, rewrite the integral, integrate, and back‑substitute. As you progress, try to group problems by the type of substitution: linear, power, trigonometric, exponential, and rational. Challenge yourself to spot the substitution in under five seconds. After completing the exercise, revisit any problem you found difficult and solve it again without looking at your earlier work. This layered practice converts procedural knowledge into automatic intuition, which is precisely what you need when facing the pressure of an IB exam.
开始时可打开笔记做前几道题，刻意写出每一步：声明 u，计算 du，调整常数，改写积分，积分，代回。随着深入，尝试按代换类型将题目分组：线性代换、幂代换、三角代换、指数代换和有理式代换。挑战自己在五秒内看出所需的代换。完成整组练习后，重新回顾任何你觉得困难的题目，不看之前的解答再解一次。这种分层训练能将程序性知识转化为自动化的直觉，而这正是在IB考试压力下你需要的能力。
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