📚 Exercise 6B.4: Integration by Substitution | 练习6B.4：换元积分法
Integration by substitution is one of the most powerful techniques for finding antiderivatives in IB Mathematics Analysis & Approaches. Exercise 6B.4 focuses on mastering this reverse chain rule method, which allows you to integrate composite functions that are not immediately obvious. This article unpacks the method step by step, explores typical problem types, and highlights common pitfalls so you can approach the exercise with confidence.
换元积分法是 IB 数学分析与方法中求原函数最有力的技巧之一。练习 6B.4 着重于掌握这种逆向链式法则的方法，它能帮助你积出那些并非一眼就能看出结果的复合函数。本文将逐步拆解该方法、探讨典型题型并指出常见陷阱，使你能够自信地完成练习。
1. The Chain Rule in Reverse | 逆向链式法则
ation and integration are inverse processes. The chain rule states that d/dx [f(g(x))] = f'(g(x)) · g' a function and its derivative can be tackled by letting u = g(x). The differential du = g'(x) dx then transforms the integral into a basic form: ∫ f'(u) du.
微分与积分互为逆运算。链式法则表明 d/dx [f(g(x))] = f'(g(x)) · g'(x)。反向运用时可识别：若被积函数含有一个函数及其导数，就可以设 u = g(x)。微分 du = g'(x) dx 便会将积分转化为基本形式 ∫ f'(u) du。
= 2x dx. The integral becomes ∫ cos u du = sin u + C = sin(x²) + C. The method essentially 'absorbs' the derivative into du.
例如，要计算 ∫ 2x · cos(x²) dx，可识别内层函数 u = x²，du = 2x dx，积分便化为 ∫ cos u du = sin u + C = sin(x²) + C。该方法实质上是将导数“吸收”进了 du。
2. Choosing the Right Substitution | 选择合适的换元
The key to success lies in choosing u wisely. Common strategies include: letting u be the inner function in a composition, the denominator of a fraction, or an expression raised to a power whose derivative also appears. Practice helps you spot the pattern instantly.
成功的关键在于明智地选择 u。常见策略包括：令 u 为复合函数的内层函数、分数的分母或一个带幂次的表达式，同时其导数也需要出现。通过练习，你能迅速识别出这种模式。
derivative of an inner function, set u equal to that inner function. For example, in ∫ x / √1 - x² ive of (1 - x²) is -2x, so a multiple of the derivative is present; we can set u = 1 - x².
若被积函数是一个乘积，其中一个是内层函数的导数，则令 u 等于该内层函数。例如，在 ∫ x / √1 - x² dx 中，(1 - x²) 的导数是 -2x，导数的倍数恰好出现；因此可设 u = 1 - x²。
3. Linear Substitutions | 线性换元
n be taken outside the integral, often turning a complicated expression into a standard one.
当被积函数具有 f(ax + b) 的形式时，简单的线性换元 u = ax + b 就很有效。因为 du = a dx，可得 dx = du / a。常数因子 1/a 可提到积分号外，从而将复杂表达式化为标准形式。
Example: ∫ e3x+2 dx. Let u = 3x+2, then du = 3 dx, so ∫ eu (du/3) = (1/3) eu + C = (1/3) e3x+2 + C. Linear substitutions are a fundamental building block.
例如：∫ e3x+2 dx。令 u = 3x+2，则 du = 3 dx，于是 ∫ eu (du/3) = (1/3) eu + C = (1/3) e3x+2 + C。线性换元是一个基础构件。
4. Substitution for Powers of Linear Expressions | 线性表达式幂次的换元
1/2) · (u⁵/5) + C = (1/10)(2x - 5)⁵ + C. This pattern appears frequently in Exercise 6B.4.
诸如 ∫ (2x - 5)⁴ dx 的积分，既可用展开法（繁琐）也可令 u = 2x - 5，得到 du = 2 dx。积分便简化为 (1/2) ∫ u⁴ du = (1/2) · (u⁵/5) + C = (1/10)(2x - 5)⁵ + C。这一模式在练习 6B.4 中频繁出现。
ain (1/a) · (uⁿ⁺¹)/(n+1) + C. In IB exams, you are often expected to perform such integrations quickly and accurately.
这个方法可以推广：对于 ∫ (ax + b)ⁿ dx，n ≠ -1，设 u = ax + b 即得 (1/a) · (uⁿ⁺¹)/(n+1) + C。在 IB 考试中，常要求能迅速而准确地完成这类积分。
5. Definite Integrals and Changing Limits | 定积分与积分限的变换
When using substitution in a definite integral, you have two options: convert everything back to x after integrating indefinite and then use the original limits, or change the limits to u-values as you go. The latter is more efficient.
在定积分中使用换元法时，有两种选择：求出不定积分后代回 x 再用原积分限，或者在换元时直接将积分限转化为 u 值。后者更为高效。
For ∫ from x=0 to x=1 of 2x √x²+1 x²+1. When x=0, u=1; when x=1, u=2. Also du = 2x dx. The integral becomes ∫ from 1 to 2 of √u du = [ (2/3) u3/2 ] from 1 to 2. This avoids back-substitution errors.
例如，∫ 从 x=0 到 x=1 的 2x √x²+1 dx，令 u = x²+1。x=0 时 u=1；x=1 时 u=2。且 du = 2x dx。积分化为 ∫ 从 1 到 2 的 √u du = [ (2/3) u3/2 ] 从 1 到 2。这就避免了回代时出错。
∫ₐᵇ f(g(x))·g'(x) dx = ∫g(a)g(b) f(u) du
6. Trigonometric Substitutions | 三角换元
he presence of a trigonometric function and its derivative (up to a constant) is the clue.
练习 6B.4 常包含如 ∫ sin³x cos x dx 的积分。设 u = sin x 则 du = cos x dx，积分化为 ∫ u³ du = (1/4) u⁴ + C。出现三角函数及其导数（至多差一个常数）即是提示。
es: ∫ tan x dx = ∫ sin x / cos x dx, let u = cos x, du = -sin x dx, yielding -∫ du/u = -ln|u| + C = -ln|cos x| + C. Recognising these patterns is vital for IB success.
其他经典例子：∫ tan x dx = ∫ sin x / cos x dx，令 u = cos x，du = -sin x dx，得到 -∫ du/u = -ln|u| + C = -ln|cos x| + C。识别这些模式对 IB 成功至关重要。
7. Exponential and Logarithmic Substitutions | 指数与对数换元
Integrals involving eᵏˣ often pair well with u = kx. For ∫ x ex² dx, set u = x², du = 2x dx, giving (1/2)∫ eu du. Similarly, for ∫ (ln x)/x dx, let u = ln x, du = (1/x) dx, yielding ∫ u du.
涉及 eᵏˣ 的积分常与 u = kx 搭配。对 ∫ x ex² dx，设 u = x²，du = 2x dx，得 (1/2)∫ eu du。类似地，对 ∫ (ln x)/x dx，令 u = ln x，du = (1/x) dx，得 ∫ u du。
These examples highlight the importance of spotting a function and its derivative. IB questions frequently test your ability to link logarithmic and exponential functions with their derivatives.
这些例子突显了识别函数及其导数的重要性。IB 题目经常考察你将对数函数和指数函数与其导数相联系的能力。
8. Worked Examples in the Style of 6B.4 | 练习 6B.4 风格的例题
Let's work through two typical problems. Example 1: du = 3x² dx, so the integral becomes (1/3) ∫ u⁵ du = (1/3)·(u⁶/6) + C = (1/18)(x³+1)⁶ + C.
让我们细解两道典型题目。例题 1：∫ x² (x³ + 1)⁵ dx。设 u = x³ + 1，du = 3x² dx，积分化为 (1/3) ∫ u⁵ du = (1/3)·(u⁶/6) + C = (1/18)(x³+1)⁶ + C。
Example 2: os 2x sin² 2x dx. Let u = sin 2x, du = 2 cos 2x dx. When x=0, u=0; when x=π/6, u=sin(π/3)=√3/2. The integral becomes (1/2) ∫ from 0 to √3/2 of u² du = (1/2)[u³/3] from 0 to √3/2 = (1/6)(3√3/8) = √3/16.
例题 2：计算 ∫ 从 0 到 π/6 的 cos 2x sin² 2x dx。令 u = sin 2x，du = 2 cos 2x dx。x=0 时 u=0；x=π/6 时 u=sin(π/3)=√3/2。积分化为 (1/2) ∫ 从 0 到 √3/2 的 u² du = (1/2)[u³/3] 从 0 到 √3/2 = (1/6)(3√3/8) = √3/16。
Always remember to adjust the constants carefully — IB mark schemes penalise missing constant multipliers heavily.
切记仔细调整常数——IB 阅卷标准对遗漏常数乘子的扣分很重。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Mistake 1: Forgetting to change the limits in definite integrals. Always convert the x-limits to u-values immediately after substitution. Mistake 2: Misidentifying the inner function or forgetting the derivative factor. Double-check by differentiating your proposed u.
错误 1：定积分中忘记变换积分限。换元后务必将 x 积分限立即转化为 u 值。错误 2：错误识别内层函数或遗忘导数因子。通过对所设 u 求导来进行复查。
Mistake 3: Incorrect handling of constants. Always write du = g'(x) dx exactly, then solve for dx if necessary. Mistake 4: Attempting substitution when it is not needed — sometimes simple algebraic manipulation is quicker.
错误 3：常数处理不当。务必准确写出 du = g'(x) dx，必要时解出 dx。错误 4：在不需要换元时强行换元——有时简单的代数变形更快。
| Error | Tip |
|---|---|
| Forgetting du/dx factor | Always write du = derivative dx |
| Not adjusting limits | Plug original x-values into u(x) immediately |
| Adding unnecessary absolute values | Check domain; only needed for logs of variable expressions |
Check: d/dx of your answer should give back the original integrand.
检验：对你的答案求导应还原为原始被积函数。
10. IB Exam Strategies and Exercise 6B.4 Application | IB 应试策略与练习 6B.4 的应用
In IB Mathematics AA exams, substitution questions often carry 4–6 marks. Show clear steps: state the substitution, write du, change limits if definite, integrate in u, and substitute back. Presentation matters.
在 IB 数学 AA 考试中，换元题通常占 4–6 分。要展示清晰步骤：写明换元、写出 du、若为定积分则变换上下限、对 u 积分、再回代。书写表达至关重要。
When tackling Exercise 6B.4, start by scanning the integrand for a function and its derivative. Underline the inner function mentally. If the structure is not obvious, try guessing u and checking du. With practice, this becomes second nature.
对付练习 6B.4 时，先快速扫视被积函数，寻找函数及其导数。在心中划出内层函数。若结构不明显，可尝试猜测 u 并检验 du。通过练习，这会成为你的第二本能。
Remember: IB questions often combine substitution with other techniques like partial fractions or trig identities, but substitution remains the first tool to try for composite functions.
请记住：IB 题目常将换元法与其他技巧（如部分分式或三角恒等式）结合，但对于复合函数，换元仍是应首先尝试的工具。
11. Practice Drills and Self-Check | 练习自测
thout solutions first: (a) ∫ (3x+2)⁹ dx, (b) ∫ cos x / sin² x dx, (c) ∫ from 0 to 2 of x ex² dx. Then check by differentiating your answers. This active recall builds fluency.
先尝试不看答案完成以下题目：(a) ∫ (3x+2)⁹ dx，(b) ∫ cos x / sin² x dx，(c) ∫ 从 0 到 2 的 x ex² dx。然后通过对答案求导来检验。这种主动回忆能培养熟练度。
After finishing Exercise 6B.4, reflect on which substitutions you found hardest. Create a mini summary card with common integral patterns and their u-substitutions. Review it before assessments.
完成练习 6B.4 后，反思你觉得最困难的换元是哪些。制作一张小卡片，总结常见积分模式及其 u-换元，在评估前进行复习。
12. Summary and Final Tips | 总结与最后的建议
Integration by substitution is a must-master skill for IB Mathematics. Always look for the inner function, compute du accurately, and handle limits carefully. The more you practice, the more intuitive these decisions become. Exercise 6B.4 is your training ground — treat it as a puzzle to unlock your calculus fluency.
换元积分法是 IB 数学必须掌握的技能。始终寻找内层函数、准确计算 du 并谨慎处理积分限。练习越多，这些决策就越直觉化。练习 6B.4 就是你的训练场——将它视为解锁微积分流利度的解谜游戏。
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