Integration by Substitution for Edexcel A-Level | 代换积分法精讲

📚 Integration by Substitution for Edexcel A-Level | 代换积分法精讲

This article revises integration by substitution, a core technique in Edexcel A-Level Mathematics, directly linked to the content found in resources like pdfjoiner(4)-199. Mastery of this method is essential for evaluating integrals that cannot be solved by standard formulas alone, especially when a function and its derivative appear together in the integrand.

本文专门梳理代换积分法，这是 Edexcel A-Level 数学的核心技巧，与 pdfjoiner(4)-199 等复习资料中的内容直接对应。掌握该方法对于求解那些无法仅靠标准公式积分的表达式至关重要，尤其是当被积函数中同时出现一个函数及其导数时。

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1. The Idea Behind Substitution | 代换法的基本思想

Integration by substitution reverses the chain rule for differentiation. If we have a composite function where the inner function’s derivative is also present, we can replace the inner function with a single variable u. This simplifies the integral into a standard form that we can evaluate directly.

代换积分法本质上是链式求导法则的逆过程。当我们面对一个复合函数，且内层函数的导数也出现在被积函数中时，就可以用一个单独的变量 u 替换内层函数。这样可以把积分化简为可直接套用公式的标准形式。

The fundamental formula capturing this idea is:

体现这一思想的基本公式为：

∫ f(g(x)) · g'(x) dx = ∫ f(u) du, where u = g(x).

Once we recognise the pattern f(g(x))·g'(x), we substitute u = g(x) and du = g'(x) dx. The integral with respect to u is often much easier to compute.

一旦识别出 f(g(x))·g'(x) 的模式，就设 u = g(x)，则 du = g'(x) dx。关于 u 的积分通常要容易计算得多。

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2. The Chain Rule in Reverse | 链式法则的逆运算

Differentiation of a composite function y = F(g(x)) gives dy/dx = F'(g(x))·g'(x). So when we see an integral of the form ∫ F'(g(x))·g'(x) dx, the antiderivative is simply F(g(x)) + C. Substitution makes this relationship explicit.

对复合函数 y = F(g(x)) 求导得到 dy/dx = F'(g(x))·g'(x)。因此，当我们看到形如 ∫ F'(g(x))·g'(x) dx 的积分时，其原函数就是 F(g(x)) + C。代换法将这一关系清晰地呈现出来。

Consider the integral ∫ 2x·cos(x²) dx. Here, the inner function is x² and its derivative 2x is present. By letting u = x², we have du = 2x dx, transforming the integral into ∫ cos u du = sin u + C = sin(x²) + C.

以积分 ∫ 2x·cos(x²) dx 为例。此处内层函数为 x²，其导数 2x 恰好出现。令 u = x²，则 du = 2x dx，原积分就化为 ∫ cos u du = sin u + C = sin(x²) + C。

The power of substitution lies in this systematic conversion: we replace the complicated x-part with a single u-variable, integrate, and then substitute back.

代换法的威力在于系统化的转换：用单一变量 u 替换复杂的 x 部分，积分后再将 u 换回原变量。

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3. Choosing an Effective Substitution | 如何选择有效的代换

Selecting the right u is the key step. In Edexcel exams, hints are often given, but you must also learn to identify patterns. Common choices include:

选择合适的 u 是关键步骤。在 Edexcel 考试中题目通常会给出提示，但你仍需学会识别常见模式。常用的代换选择包括：

• u = expression inside a bracket raised to a power, e.g., u = 2x+1 in ∫ (2x+1)⁵ dx. | u = 带幂次的括号内的表达式，如 ∫ (2x+1)⁵ dx 中设 u = 2x+1。
• u = denominator of a rational function, e.g., u = x²+3 in ∫ 6x/(x²+3) dx. | u = 有理函数的分母，如 ∫ 6x/(x²+3) dx 中设 u = x²+3。
• u = the argument of a trigonometric function, e.g., u = 3x in ∫ sin 3x dx. | u = 三角函数的变量部分，如 ∫ sin 3x dx 中设 u = 3x。
• u = an exponential exponent, e.g., u = x² in ∫ x·e^(x²) dx. | u = 指数函数的指数部分，如 ∫ x·e^(x²) dx 中设 u = x²。
• u = a logarithmic function, e.g., u = ln x in ∫ (ln x)/x dx. | u = 对数函数本身，如 ∫ (ln x)/x dx 中设 u = ln x。

Always check that the derivative du/dx or a constant multiple of it appears elsewhere in the integrand. This confirms your choice is workable.

始终要检查 du/dx 或其常数倍是否出现在被积函数的其他地方。这一检验能确认代换是否可行。

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4. Handling Definite Integrals | 处理定积分

When working with definite integrals, substitution requires an extra step: changing the limits. If the original integral is ∫_a^b f(x) dx and we substitute u = g(x), the limits become u = g(a) and u = g(b). There is no need to convert back to x after integration.

处理定积分时，代换需要额外一步：变换积分上下限。若原积分为 ∫_a^b f(x) dx，代换 u = g(x) 后，积分限变为 u = g(a) 和 u = g(b)。完成对 u 的积分后无需再将 u 换回 x。

For example, evaluate ∫₀¹ 2x√(1+x²) dx. Let u = 1+x². Then du = 2x dx. When x=0, u=1; when x=1, u=2. The integral becomes ∫₁² √u du = [ (2/3)u^(3/2) ]₁² = (2/3)(2√2 − 1). This method avoids reverting to x and saves time.

例如计算 ∫₀¹ 2x√(1+x²) dx。令 u = 1+x²，则 du = 2x dx。当 x=0 时 u=1；x=1 时 u=2。积分化为 ∫₁² √u du = [ (2/3)u^(3/2) ]₁² = (2/3)(2√2 − 1)。此法无需回到变量 x，能有效节省时间。

Always rewrite the limits immediately after the substitution to prevent mistakes. A common error is to keep the original x-limits while integrating with respect to u.

代换后务必立即重写积分限，以免出错。常见错误就是在对 u 积分时仍保留原来的 x 积分限。

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5. Integrating by Substitution with Trigonometric Functions | 三角函数的代换积分

Trigonometric integrands frequently require substitution, especially when powers of sine and cosine are involved. The general strategy is to let u equal the trigonometric function whose derivative is present, or to use an identity to simplify first.

含有三角函数的被积函数经常需要代换，尤其是在出现正弦或余弦的幂次时。一般策略是令 u 等于某个三角函数，前提是其导数也出现在积分中，或者先用三角恒等式化简。

Consider ∫ sin³x cos x dx. Here the derivative of sin x is cos x. Let u = sin x, then du = cos x dx, giving ∫ u³ du = (1/4)u⁴ + C = (1/4)sin⁴x + C. This is a classic Edexcel P4 example.

考虑 ∫ sin³x cos x dx。这里 sin x 的导数 cos x 恰好存在。令 u = sin x，则 du = cos x dx，积分变为 ∫ u³ du = (1/4)u⁴ + C = (1/4)sin⁴x + C。这是一个典型的 Edexcel P4 考题。

For integrals like ∫ sin⁵x dx, where the derivative is not fully present, we often separate one factor and use the identity sin²x = 1−cos²x, then substitute u = cos x. The choice depends on the available derivative.

对于形如 ∫ sin⁵x dx 的积分，其导数并未完全出现。此时通常分离出一个因式，并利用恒等式 sin²x = 1−cos²x，再代换 u = cos x。选择取决于哪种导数更易得到。

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6. Substitution with Exponential and Logarithmic Functions | 指数函数与对数函数的代换

Exponential integrands of the form ∫ f'(x)·e^(f(x)) dx are natural candidates for u = f(x). The derivative f'(x) is the key signal. Similarly, when a 1/x term accompanies ln x, we can set u = ln x.

形如 ∫ f'(x)·e^(f(x)) dx 的指数型被积函数是代换 u = f(x) 的天然对象，其中导数 f'(x) 是关键信号。类似地，若 1/x 项伴随 ln x 出现，则可设 u = ln x。

Example: ∫ x·e^(x²) dx. Let u = x², du = 2x dx, so the integral becomes (1/2)∫ e^u du = (1/2)e^(x²) + C. The constant factor 1/2 must be handled carefully.

例题：∫ x·e^(x²) dx。令 u = x²，du = 2x dx，于是积分化为 (1/2)∫ e^u du = (1/2)e^(x²) + C。常数因子 1/2 必须谨慎处理。

For ∫ (ln x)/x dx, the substitution u = ln x, du = (1/x) dx works perfectly, giving ∫ u du = (1/2)(ln x)² + C. Recognising these patterns comes with practice.

对于 ∫ (ln x)/x dx，代换 u = ln x，du = (1/x) dx 完美适用，得到 ∫ u du = (1/2)(ln x)² + C。通过练习，识别这些模式将变得轻松。

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7. Special Case: Linear Substitutions | 特殊情形：线性代换

When the inner function is linear, i.e., of the form u = ax+b, the substitution simplifies considerably because du = a dx. This is essentially the reverse of integrating a function of a linear expression, and the formula is sometimes given in formula booklets.

当内层函数为线性表达式，即形如 u = ax+b 时，代换法大大简化，因为 du = a dx。这本质上是求线性表达式的函数的积分的逆过程，公式有时会直接出现在公式表中。

For instance, ∫ e^(3x+1) dx can be solved by setting u = 3x+1, du = 3 dx, so dx = du/3. The integral becomes (1/3)∫ e^u du = (1/3)e^(3x+1) + C. This is faster than remembering the standard result ∫ e^(ax+b) dx = (1/a)e^(ax+b) + C.

例如 ∫ e^(3x+1) dx 可设 u = 3x+1，du = 3 dx，即 dx = du/3。积分变为 (1/3)∫ e^u du = (1/3)e^(3x+1) + C。这比直接记忆公式 ∫ e^(ax+b) dx = (1/a)e^(ax+b) + C 更加直观。

In exams, you may be asked to use a given linear substitution. Always show the step du = a dx and adjust the integral accordingly.

考试中可能会要求使用某个给定的线性代换。记得明确写出 du = a dx 这一步，并相应调整积分。

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8. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Even strong candidates can slip up with substitution. Watch out for these errors:

即使是能力出众的考生也可能在代换中失误。请警惕以下常见错误：

• Forgetting to change limits in definite integrals. Always update the limits immediately after the substitution. | 忘记变换定积分的上下限。代换后务必立即更新积分限。
• Misplacing the derivative factor. If du = 2x dx, then x dx = du/2, not du. Missing the factor leads to an incorrect answer. | 导数因子处理不当。若 du = 2x dx，则 x dx = du/2，而非 du。漏掉该因子会导致答案错误。
• Not expressing the entire integrand in u. All x terms and dx must be replaced. A stray x left behind makes the integral unsolvable. | 未将整个被积函数用 u 表示。所有 x 项及 dx 都必须替换。残留的 x 会让积分无法求解。
• Ignoring the need to substitute back for indefinite integrals. The final answer must be given in terms of the original variable x. | 不定积分忘记回代原变量。最终答案必须以原变量 x 给出。
• Overcomplicating the choice of u. If a simple u does not work, look for a cosine-sine pairing or a bracket-power pattern before trying something exotic. | 代换 u 的选择过于复杂。若简单代换无效，先尝试余正弦配对或括号幂次模式，而非立刻选择怪异函数。

Checking by differentiation is an excellent habit. If you differentiate your answer and obtain the original integrand, the substitution was applied correctly.

求导验算是一个极好的习惯。若对答案求导后得出原被积函数，则代换过程正确无误。

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9. Exam-Style Questions and Strategies | 考试真题解析与策略

Edexcel A-Level papers often embed substitution within larger problems, such as finding areas under curves or solving differential equations. A typical question might be: ‘Use the substitution u = x²+1 to find ∫ x/(x²+1) dx.’

Edexcel A-Level 试卷常将代换法嵌入更大的问题中，如求曲线下方面积或解微分方程。一道典型题目可能是：“使用代换 u = x²+1 求 ∫ x/(x²+1) dx。”

Solution approach: Set u = x²+1, then du = 2x dx → x dx = du/2. The integral becomes (1/2)∫ 1/u du = (1/2) ln|u| + C = (1/2) ln(x²+1) + C. For definite integrals, use u-limits and evaluate with exact values.

解法思路：令 u = x²+1，则 du = 2x dx → x dx = du/2。积分变为 (1/2)∫ 1/u du = (1/2) ln|u| + C = (1/2) ln(x²+1) + C。对于定积分，使用 u 限并给出精确值。

Another common question involves trigonometric substitution, such as ∫ sin 2x cos³ 2x dx. Let u = cos 2x, du = −2 sin 2x dx. The integral becomes −(1/2)∫ u³ du = −(1/8)u⁴ + C = −(1/8)cos⁴ 2x + C. Pay close attention to the negative sign.

另一常见题型涉及三角代换，如 ∫ sin 2x cos³ 2x dx。令 u = cos 2x，du = −2 sin 2x dx。积分变为 −(1/2)∫ u³ du = −(1/8)u⁴ + C = −(1/8)cos⁴ 2x + C。要格外留意负号。

When the substitution is not given, the key is to look for a function and its derivative. If the integrand is a fraction where the numerator is the derivative of the denominator, u = denominator is usually effective.

当题目未给出代换时，关键就在寻找函数及其导数的配对。若被积函数为分式，且分子恰好是分母的导数，那么设 u = 分母往往有效。

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10. Summary and Key Takeaways | 总结与核心要点

Integration by substitution is one of the most powerful tools in the Edexcel A-Level armoury. Its success depends on pattern recognition and systematic execution.

代换积分法是 Edexcel A-Level 数学中最强大的工具之一，其成功运用取决于模式识别和系统性操作。

Remember the mantra: Choose u, find du, replace all x and dx, integrate in u, and either convert limits or substitute back. With sufficient practice using resources like pdfjoiner(4)-199, the process becomes almost automatic.

记住这一准则：选取 u，求出 du，替换所有 x 和 dx，对 u 积分，然后要么转化积分限，要么回代原变量。通过结合 pdfjoiner(4)-199 等资源的充分练习，这个过程将变得近乎本能。

Master this technique, and you will handle a wide range of integration problems with confidence, from simple polynomial composites to tricky trigonometric and exponential functions. Always check your work by differentiating the result.

熟练掌握该技巧后，你将能自信地应对各类积分问题，从简单的多项式复合函数到棘手的三角与指数函数。务必通过求导结果来检查答案。

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