📚 Integration by Substitution | 换元积分法
Integration by substitution is one of the most powerful techniques in the Edexcel A-Level Mathematics syllabus. It is essentially the reverse of the chain rule for differentiation and allows us to tackle integrals that, at first glance, look complicated. By introducing a new variable, we can transform an intimidating integral into a standard form that is easy to evaluate. Mastering this technique is essential for success in Pure Mathematics papers, where it routinely appears in questions on indefinite integrals, definite integrals, and differential equations.
换元积分法是 Edexcel A-Level 数学大纲中最强大的积分技巧之一。它本质上是微分链式法则的逆运算，能够帮助我们处理那些初看十分复杂的积分。通过引入新变量，可以将一个令人生畏的积分转化为容易计算的标准形式。掌握这一技巧对于在纯数学考试中取得成功至关重要，因为它经常出现在不定积分、定积分以及微分方程的题目中。
1. The Concept of Integration by Substitution | 换元积分法的概念
rule tells us that if we differentiate a composite function y = f(g(x)), we get dy/dx = f'(g(x))·g' oduct of a function of g(x) and the derivative of g(x), we can set u = g(x) so that du = g'(x) dx. The integral then simplifies to ∫ f'(u) du, which is much easier to evaluate. This technique is sometimes called the 'reverse chain rule' or 'u-substitution'.
链式法则告诉我们，如果对复合函数 y = f(g(x)) 求导，得到 dy/dx = f'(g(x))·g'(x)。将这个思路逆转过来就得到了换元法。如果被积函数可以写成某个关于 g(x) 的函数与 g(x) 的导数之积，我们就可以令 u = g(x)，从而 du = g'(x) dx。此时积分简化为 ∫ f'(u) du，这易于计算。这个技巧有时被称为“反向链式法则”或“u代换”。
2. Recognizing When to Use Substitution | 何时使用换元法
r. For example, in ∫ 2x·cos(x²) dx, the derivative of x² is 2x, which sits right next to cos(x²). Other common patterns include integrands of the form f(ax+b), where the inner function is linear, and integrands involving roots of quadratics or rational expressions where the numerator is nearly the derivative of the denominator.
并不是每个积分都需要换元。一个明显的信号是，当你看到某个函数与其导数同时出现时，换元法就会派上用场。例如，在 ∫ 2x·cos(x²) dx 中，x² 的导数正是旁边的 2x，而 cos(x²) 是函数本身。其他常见的模式还包括被积函数形如 f(ax+b)（内层函数为线性）、含有二次根式的积分，或者分子恰好是分母导数的倍数等情况。
3. Basic Substitution Steps | 基本换元步骤
Step 1: hat will simplify the integral. Often, this is the inner function of a composite function.
Step 2: Differentiate u with respect to x to obtain du/dx = g'(x), then rearrange to express dx in terms of du: dx = du / g'(x).
Step 3: Replace all instances of x and dx in the original integral with expressions in u, aiming to cancel any remaining x terms. The resulting integral should be a standard form in u only.
Step 4: Integrate with respect to u.
Step 5: = g(x) to express the answer in terms of the original variable (for indefinite integrals).
步骤 1：选取能够简化积分的代换 u = g(x)。通常，它就是复合函数的内层函数。
步骤 2：对 u 关于 x 求导，得到 du/dx = g'(x)，然后变形为 dx = du / g'(x)。
步骤 3：将原积分中所有的 x 和 dx 均用 u 的表达式替换，尽量消去剩余的所有 x 项，使得新积分仅含 u 的标准形式。
步骤 4：对 u 进行积分。
步骤 5：回代 u = g(x)，将答案表示为原变量的函数（适用于不定积分）。
4. Example: Integrating x√x²+1 | 例题：积分 x√x²+1
Let us work through a classic example that appears frequently in Edexcel papers.
Solution: itution u = x²+1. Then du/dx = 2x, which gives dx = du/(2x). Substitute into the integral:
∫ x√x²+1 dx = ∫ x·√u · (du/(2x)) = ∫ (1/2) √u du.
The x factors cancel perfectly. Now integrate: ∫ (1/2) u½ du = (1/2)·(2/3) u3/2 + C = (1/3) u3/2 + C. Finally, substitute back u = x²+1 to obtain (1/3)(x²+1)3/2 + C. This example shows the power of a well-chosen substitution.
我们来完成一个在 Edexcel 试卷中经常出现的经典例题。
解答：注意到 x²+1 的导数为 2x，而分子恰好含有 x。令 u = x²+1，则 du/dx = 2x，即 dx = du/(2x)。代入积分：
∫ x√x²+1 dx = ∫ x·√u · (du/(2x)) = ∫ (1/2) √u du.
x 项完美消去。现在积分：∫ (1/2) u½ du = (1/2)·(2/3) u3/2 + C = (1/3) u3/2 + C。最后回代 u = x²+1，得到 (1/3)(x²+1)3/2 + C。这个例子展示了一个恰当代换的强大之处。
5. Trigonometric Substitutions | 三角换元
When integrands contain expressions like √a² - x², √a² + x² or √x² - a², trigonometric substitutions are extremely effective. For instance, for √a² - x², use x = a sin θ, then dx = a cos θ dθ and √a² - x² becomes a cos θ. This transforms the integral into a trigonometric one, which can be solved using standard trig integrals. After integrating, you must convert back to x using inverse trigonometric functions or a right-angled triangle.
当被积函数含有形如 √a² - x²、√a² + x² 或 √x² - a² 的部分时，三角代换非常有效。例如，对于 √a² - x²，令 x = a sin θ，则 dx = a cos θ dθ，且 √a² - x² 变为 a cos θ。这样就把积分转化为三角函数的积分，随后可以使用标准三角积分公式求解。积分完成后，需利用反三角函数或直角三角形将结果换回 x 的表达形式。
6. Definite Integrals and Changing Limits | 定积分与换限
f(x) dx, and you set u = g(x). Then the new limits become u₁ = g(a) and u₂ = g(b). The integral transforms to ∫(from u₁ to u₂) f(g⁻¹(u))·(dx/du) du, but we usually write it directly as ∫(from u₁ to u₂) (something) du. This method saves time and reduces errors. Always adjust the limits immediately after determining the substitution.
对于定积分，我们可以在换元的同时改变积分上下限，从而免去回代步骤。假设 ∫(从 a 到 b) f(x) dx，令 u = g(x)，则新的积分限变为 u₁ = g(a) 和 u₂ = g(b)。积分转化为 ∫(从 u₁ 到 u₂) 关于 u 的被积函数 du。这样可以节省时间并减少差错。务必在确定代换后立刻调整积分限。
7. Using Substitution with Exponential Functions | 指数函数换元
Exponential integrals often benefit from substitution. Consider ∫ ex² ch appears as a multiplier. Let u = x², du/dx = 2x, dx = du/(2x). The integral becomes ∫ eu du = eu + C = ex² + C. Without substitution, this integral looks difficult, but it collapses into the simplest exponential integral. Watch out for such patterns in exam questions.
涉及指数函数的积分常常通过代换得以简化。考虑 ∫ ex²·2x dx，其中 x² 的导数 2x 正好作为乘数出现。令 u = x²，du/dx = 2x，dx = du/(2x)，积分化为 ∫ eu du = eu + C = ex² + C。若不使用代换，这个积分看似困难，但实际上退化为最简单的指数积分。在考试中要留心类似模式。
8. Handling Rational Functions | 有理函数换元
A common Edexcel trick is an integral of the form ∫ f'(x)/f(x) dx, which directly gives ln|f(x)| able, but ∫ (x+1)/(x²+2x+5) dx might require substitution u = x²+2x+5, because du = (2x+2) dx = 2(x+1) dx. Half the numerator appears, so we adjust constants and integrate to get a natural logarithm. More complex rational functions may require splitting or partial fractions before substitution.
Edexcel 考试中常见的技巧是形如 ∫ f'(x)/f(x) dx 的积分，可直接得出 ln|f(x)| + C。然而，许多有理函数积分可通过巧妙代换转化为这种形式。例如，∫ (2x)/(x²+3) dx 一眼就能认出，但 ∫ (x+1)/(x²+2x+5) dx 可能需要令 u = x²+2x+5，因为 du = (2x+2) dx = 2(x+1) dx。分子的一半出现，于是调整常数并积分得到自然对数。更复杂的有理函数可能需要在代换前进行拆分或部分分式分解。
9. Common Mistakes | 常见错误
Forgetting to replace dx: Many students substitute u but leave dx unchanged, leading to a meaningless mix of variables. Always compute du/dx and express dx in terms of du and the new variable.
Not cancelling all x terms: The substitution should eliminate the original variable completely. If x remains after substitution, either the substitution is wrong or an algebraic error has been made.
Incorrect limits in definite integrals: he : If you change the variable but keep the original limits, your final answer will be wrong. Always convert limits using u = g(x).
Losing constants: Adjust constants carefully when substituting; missing a factor of 2 is a frequent source of error.
忘记替换 dx：许多学生做了代换却让 dx 原封不动，导致变量混杂。务必计算出 du/dx 并用 du 和新变量表示 dx。
未能消去所有 x 项：代换理应完全消去原变量。如果代换后仍有 x 残留，要么代换选择错误，要么出现了代数运算失误。
定积分上下限错误：如果换了变量却保留了原积分限，最终答案必然错误。一定要使用 u = g(x) 换算积分限。
丢失常数：代换时需仔细调整常数；漏掉一个因子 2 是常见的错误来源。
10. Exam Tips for Edexcel | Edexcel 考试技巧
Edexcel examiners typically allocate 6–9 marks for a substitution integration question. Show every step clearly: state your substitution, find du/dx, rewrite the integral, perform the integration, and adjust limits if it is definite. Even if you make a mistake in the algebra, you can earn method marks. Practice recognising substrates that are given in the question—sometimes the substitution is provided, so use it exactly as instructed. Also, be comfortable with both indefinite and definite forms, and know how to use trigonometric identities when substituting with trig functions. Finally, check your answer by differentiating; in an exam, this can save you from losing easy marks.
Edexcel 考官通常为一道换元积分题分配 6–9 分。务必清晰展示每一步：写出代换式，求 du/dx，重写积分，执行积分，如果是定积分还需调整积分限。即使代数运算有误，你仍可获得方法分。平时要多练习识别题目中给出的代换——有时试卷会直接提供代换式，你要严格按指令使用。此外，要熟悉不定积分和定积分两种形式，并掌握用三角恒等式处理三角代换的技巧。最后，用求导检查答案；在考试中这能帮你避免丢掉容易得到的分数。
Published by TutorHao | 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