Exercise 17F: Integration by Substitution | 练习17F：换元积分法

📚 Exercise 17F: Integration by Substitution | 练习17F：换元积分法

Exercise 17F in your IB Mathematics course focuses on one of the most essential techniques for finding antiderivatives: integration by substitution. This method, often called u-substitution, allows you to transform a complicated integral into a basic one by changing the variable. Mastering this technique is crucial for success in both Analysis & Approaches (AA) and Applications & Interpretation (AI) courses, as it appears across calculus topics from simple polynomials to trigonometric and exponential functions.

IB数学课程中的练习17F专注于求不定积分最重要的技巧之一：换元积分法。这种方法通常称为 u 代换，它通过更换变量，将一个复杂的积分转化为基本积分。掌握这一技巧对于在分析与方法（AA）以及应用与解释（AI）两门课程中取得成功至关重要，因为它遍及从简单多项式到三角和指数函数的整个微积分主题。

1. The Core Idea of Exercise 17F | 练习17F的核心思想

Exercise 17F is designed to build fluency in recognising when an integrand is the result of the chain rule. Instead of expanding or simplifying, you reverse the differentiation process by letting u be the inner function. This turns a product or composite expression into something directly integrable.

练习17F旨在培养你识别被积函数何时为链式法则结果的流畅度。你需要通过令 u 等于内层函数来逆转微分过程，而不是展开或化简。这样就能将一个乘积或复合表达式转化为可以直接积分的形式。

The problems in this exercise gradually increase in difficulty, starting from straightforward linear inner functions and moving towards trigonometric, exponential, and rational combinations. You will also practise handling definite integrals by adjusting the limits of integration.

本练习中的题目难度逐步递增，从简单的线性内层函数开始，逐步过渡到三角、指数和有理式的组合。你还将通过调整积分限来练习处理定积分。

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2. Recognising the Pattern: Reverse Chain Rule | 识别模式：逆链式法则

If you see an integral of the form ∫ f ‘ (g(x)) · g ‘ (x) dx, you can immediately apply the reversal of the chain rule. In practice, this often means spotting a function and its derivative appearing together. For example, ∫ 2x cos(x²) dx contains x² and the derivative 2x, making it a perfect candidate for substitution.

如果你看到形如 ∫ f ‘ (g(x)) · g ‘ (x) dx 的积分，就可以直接应用链式法则的逆运算。在实际操作中，这通常意味着要发现一个函数和它的导数同时出现。例如，∫ 2x cos(x²) dx 包含了 x² 及其导数 2x，这便是一个完美的代换候选。

When working through Exercise 17F, always scan the integrand for a factor that looks like the derivative of another part. This skill is tested repeatedly in both algebraic and transcendental functions.

在做练习17F时，一定要扫视被积函数，寻找一个因子是否看起来像是另一部分的导数。这一技能在代数函数和超越函数中都会被反复考查。

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3. Choosing the Substitution u | 选择代换变量 u

A systematic way to begin is letting u = g(x), where g(x) is the ‘inside’ function of a composition. Then compute du = g ‘ (x) dx. The goal is to replace every x-term with an expression in u and obtain an integral purely in terms of u. If any x remains after the substitution, you have chosen the wrong u or need to rewrite x in terms of u.

一个系统的方法是设 u = g(x)，其中 g(x) 是复合函数中的“内层”函数，然后计算 du = g ‘ (x) dx。目标是将每一个含 x 的项替换为关于 u 的表达式，并得到一个仅含 u 的积分。如果代换后仍有任何 x 残留，要么你选错了 u，要么需要再将 x 用 u 表示出来。

In Exercise 17F, you will often see substitutions like u = 3x + 5, u = x² + 1, or u = sin x. These simple choices drastically simplify the integral and prepare you for more advanced IB exam problems.

在练习17F中，你会经常看到诸如 u = 3x + 5, u = x² + 1 或 u = sin x 的代换。这些简单的选择会极大地简化积分，并为你解决更高级的IB考题做好准备。

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4. Substitution with Linear Inner Functions | 线性内层函数的代换

When the inner function is linear, such as u = 2x + 3, we have du = 2 dx, so dx = ½ du. This allows us to replace dx directly. For instance, ∫ (2x+3)⁵ dx becomes ½ ∫ u⁵ du. This is often the first type of problem in Exercise 17F, building confidence in the mechanical steps.

当内层函数是线性函数时，例如 u = 2x + 3，我们有 du = 2 dx，因此 dx = ½ du。这使我们能够直接替换 dx。例如，∫ (2x+3)⁵ dx 变为 ½ ∫ u⁵ du。这通常是练习17F中的第一种题型，有助于建立对机械步骤的信心。

Remember to multiply by the reciprocal of the coefficient from du/dx when replacing dx. A common mistake in Exercise 17F is forgetting this constant factor, which leads to an incorrect antiderivative by a multiple.

在替换 dx 时，记得乘上 du/dx 系数的倒数。练习17F中一个常见错误就是忘记这个常数因子，从而导致不定积分结果相差一个倍数。

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5. Worked Basic Example | 基础例题演算

Consider ∫ 6x² (x³ + 4)⁸ dx. Let u = x³ + 4, then du = 3x² dx, so 6x² dx = 2 du. The integral becomes ∫ 2 u⁸ du = (2/9) u⁹ + C = (2/9)(x³ + 4)⁹ + C. This method avoids expanding the eighth power, saving time and reducing errors.

考虑 ∫ 6x² (x³ + 4)⁸ dx。令 u = x³ + 4，则 du = 3x² dx，因此 6x² dx = 2 du。该积分变为 ∫ 2 u⁸ du = (2/9) u⁹ + C = (2/9)(x³ + 4)⁹ + C。这种方法避免了展开八次方的繁琐计算，节省时间并减少错误。

When writing your solution for Exercise 17F, always show the step where you express du and adjust for the constant factor. Even if you can do it mentally, explicit working helps examiners follow your logic and can earn method marks.

在书写练习17F的解答时，务必要展示你写出 du 并调整常数因子的步骤。即使你能心算，明确的步骤也能帮助阅卷人跟上你的推理，并可能获得方法分。

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6. Definite Integrals by Substitution | 用代换法求解定积分

For definite integrals, substitution requires an extra step: changing the limits. If u = g(x), then when x = a, u = g(a); when x = b, u = g(b). You can then evaluate the definite integral entirely in u without converting back to x. This approach is heavily examined in IB and appears in Exercise 17F with various function types.

对于定积分，代换需要额外的步骤：更换积分限。如果 u = g(x)，那么当 x = a 时，u = g(a)；当 x = b 时，u = g(b)。然后你可以在 u 的世界里完全求出定积分值，而不需转换回 x。这种方法在IB考试中经常考查，并在练习17F中搭配多种函数类型出现。

For example, evaluate ∫₀¹ 4x√(2x²+1) dx. Let u = 2x²+1, du = 4x dx; when x=0, u=1; when x=1, u=3. The integral becomes ∫₁³ √u du = [ (2/3) u^(3/2) ]₁³ = (2/3)(√27 – 1) = 2√3 – 2/3.

例如，计算 ∫₀¹ 4x√(2x²+1) dx。令 u = 2x²+1，du = 4x dx；当 x=0 时 u=1；当 x=1 时 u=3。积分变为 ∫₁³ √u du = [ (2/3) u^(3/2) ]₁³ = (2/3)(√27 – 1) = 2√3 – 2/3。

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7. Trigonometric Substitutions | 三角代换

Exercise 17F also extends substitution to trigonometric integrals. If you encounter ∫ sin⁵x cos x dx, let u = sin x, then du = cos x dx. The integral becomes ∫ u⁵ du = (1/6) sin⁶ x + C. This is much simpler than using reduction formulae or identities.

练习17F还将代换法扩展到三角积分。若遇到 ∫ sin⁵x cos x dx，令 u = sin x，则 du = cos x dx。积分变为 ∫ u⁵ du = (1/6) sin⁶ x + C。这比使用降次公式或三角恒等式简单得多。

Be careful with trigonometric functions: the derivative of cos x is –sin x, so signs must be adjusted. For instance, ∫ cos³x sin x dx could be handled by u = cos x, giving du = –sin x dx, so sin x dx = –du, resulting in –∫ u³ du.

处理三角函数时要小心：cos x 的导数是 –sin x，所以必须调整符号。例如，∫ cos³x sin x dx 可设 u = cos x，得 du = –sin x dx，因此 sin x dx = –du，从而得到 –∫ u³ du。

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8. Exponential and Rational Combinations | 指数与有理式组合

Integrals like ∫ eˣ / (1 + eˣ) dx appear frequently in Exercise 17F. The natural choice is u = 1 + eˣ, giving du = eˣ dx. The integral reduces to ∫ 1/u du = ln |u| + C = ln(1+eˣ) + C. Observing that the numerator is the derivative of the denominator is a key pattern in IB.

诸如 ∫ eˣ / (1 + eˣ) dx 的积分在练习17F中频繁出现。自然的选择是 u = 1 + eˣ，得 du = eˣ dx。积分简化为 ∫ 1/u du = ln |u| + C = ln(1+eˣ) + C。观察到分子是分母的导数，这是IB考试中的一个关键模式。

Similarly, for rational functions like ∫ x / (x²+9) dx, set u = x²+9, du = 2x dx, leading to ½ ∫ 1/u du = ½ ln(x²+9) + C. These problems reinforce the connection between logarithmic differentiation and integration.

类似地，对于有理函数如 ∫ x / (x²+9) dx，设 u = x²+9，du = 2x dx，得到 ½ ∫ 1/u du = ½ ln(x²+9) + C。这些题目强化了对数微分与积分之间的联系。

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9. Common Pitfalls in Exercise 17F | 练习17F中的常见陷阱

One frequent error is forgetting to replace dx completely. Students sometimes leave dx as it is and write du inside the integral, resulting in a meaningless mix of variables. Always explicitly write dx = … du based on du/dx. Another mistake is using a substitution that does not eliminate all x-terms, leaving an integral in both u and x which cannot be evaluated.

一个常见错误是忘记完全替换 dx。学生有时会原样保留 dx，而在积分中直接写 du，导致变量混乱无意义。务必根据 du/dx 明确地写出 dx = … du。另一个错误是使用的代换未能消去所有含 x 的项，留下的积分同时含有 u 和 x，无法求解。

In definite integrals, forgetting to change the limits is a serious error that will lead to an incorrect answer. If you write the new integral with the old x-limits, the final numerical value will be wrong. Make it a habit to write ‘when x = …, u = …’ as soon as you choose u.

在定积分中，忘记更换积分限是一个严重错误，会导致答案错误。如果你用旧的 x 限写出新的积分，最终的数值将是错误的。养成一选定 u 就写出“当 x = … 时，u = …”的习惯。

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10. Practice and Exam Strategy | 练习与应试策略

To master Exercise 17F, practise a wide range of integrals daily. Begin with linear substitution, then move to polynomials, trigonometric, exponential, and rational forms. Use past IB paper questions, as the exam often mixes substitution with areas, volumes, or kinematics, requiring you to apply the technique in context.

要精通练习17F，每天练习各种积分。从线性代换开始，然后过渡到多项式、三角、指数和有理形式。使用历年IB真题，因为考试经常将代换法与面积、体积或运动学结合，要求你在情境中应用这一技巧。

A solid exam tip: if you get stuck, check whether you can differentiate your answer to see if it yields the original integrand. This verification can catch algebraic slips. Also, remember that while substitition is powerful, some integrals require other techniques, so always assess the structure first.

一个实用的应试技巧：如果卡住了，检查你是否能对答案求导，看是否会得到原来的被积函数。这一验证能发现代数上的小错误。此外，请记住虽然代换法功能强大，但有些积分需要其他技巧，因此务必先评估结构。

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