📚 Mastering Integration with Partial Fractions | 掌握部分分式积分法
Integration with partial fractions is one of the most powerful algebraic techniques required for Edexcel A-Level Mathematics. It transforms complicated rational expressions into sums of simpler fractions, each of which can be integrated directly using standard results. This method appears regularly in the Pure Mathematics papers, often linked to differential equations, area under a curve, or parametric integration problems. Understanding how to decompose a rational function correctly and then integrate term by term is essential for achieving top marks.
部分分式积分法是 Edexcel A-Level 数学考试中最重要的代数技巧之一。它能将复杂的有理式拆分成几个简单分式之和，每个简单分式都可直接利用标准公式积分。这种方法频繁出现在纯数试卷中，常与微分方程、曲线下面积或参数方程积分等问题联合考查。正确分解有理函数并逐项积分，是获取高分的关键。
1. What is Partial Fraction Decomposition? | 什么是部分分式分解？
Partial fraction decomposition is the process of expressing a single rational function as a sum of two or more simpler rational expressions. It works when the degree of the numerator is strictly less than the degree of the denominator. If this is not the case, polynomial long division must be performed first to obtain a proper fraction. The technique is built on the idea of reversing the addition of algebraic fractions, allowing us to integrate expressions like (3x+5)/(x²+3x+2) by splitting them into terms like A/(x+1) + B/(x+2).
部分分式分解是把单个有理函数表示为两个或多个更简单有理式的和的过程。它适用于分子的次数严格低于分母次数的情况。如果不符合，则需要先用多项式长除法将其化为真分式。这项技术的核心是逆向代数分式的加法运算，使得我们可以将 (3x+5)/(x²+3x+2) 拆分成 A/(x+1) + B/(x+2) 这样的形式，从而进行积分。
The general approach relies on factorising the denominator into linear and irreducible quadratic factors. Each type of factor gives a specific form for the partial fraction. For Edexcel A-Level, you will mainly encounter distinct linear factors, repeated linear factors, and occasionally a quadratic factor that cannot be factorised further. Mastering these three cases covers almost all exam questions.
一般方法是先将分母因式分解为线性因子和不可约二次因子。每种因子对应特定的部分分式形式。对于 Edexcel A-Level 考试，你主要会遇到不重复的线性因子、重复线性因子，偶尔也会出现无法继续分解的二次因子。掌握这三种情况几乎能应对所有考题。
2. The Standard Integration Formulas You Need | 你需要牢记的标准积分公式
Before tackling partial fractions, you must be confident with the fundamental logarithmic and inverse tangent integrals. For a linear denominator (ax+b), the integral is (1/a) ln|ax+b| + C. For an irreducible quadratic denominator of the form x²+a², the integral becomes (1/a) arctan(x/a) + C. These results form the building blocks once the rational function has been decomposed.
在处理部分分式前，你必须熟练掌握基本的对数积分和反正切积分公式。对于线性分母 (ax+b)，积分结果为 (1/a) ln|ax+b| + C。对于形如 x²+a² 的不可约二次分母，积分结果为 (1/a) arctan(x/a) + C。这些结果是分解有理函数后积分的基本构件。
The Edexcel formula booklet does not provide the general partial fractions decomposition; you are expected to recall the method. However, it does include the integrals of 1/(x²+a²) and 1/(x²-a²), so learning how to manipulate fractions to match these forms is vital. Never forget the absolute value inside the logarithm unless the domain guarantees the expression is positive, but in indefinite integrals it is safest to include it.
Edexcel 的公式手册不提供一般的部分分式分解方法，你需要自己记住。不过，它确实提供了 1/(x²+a²) 和 1/(x²-a²) 的积分，因此学会将分式转化为这些标准形式至关重要。除非定义域确保表达式恒正，否则对数内绝不要遗漏绝对值符号，在不定积分中最稳妥的做法是始终保留。
3. Distinct Linear Factors | 不重复的线性因子
When the denominator factorises into distinct linear factors, the decomposition takes the form A/(x-p) + B/(x-q) + …. For example, to integrate (2x+1)/[(x-1)(x+2)], we write it as A/(x-1) + B/(x+2). Multiply through by the denominator, equate coefficients of powers of x, and solve for A and B. Then integrate each term individually to get A ln|x-1| + B ln|x+2| + C.
当分母可分解为不重复的线性因子时，分解形式为 A/(x-p) + B/(x-q) + …。例如，要对 (2x+1)/[(x-1)(x+2)] 积分，我们设其为 A/(x-1) + B/(x+2)。两边同乘以分母，比较 x 各次幂的系数，解出 A 和 B。然后逐项积分，得到 A ln|x-1| + B ln|x+2| + C。
A quicker method for linear factors is the cover-up rule: to find A, cover up (x-1) in the denominator and substitute x=1 into the remaining expression; similarly, cover up (x+2) and substitute x=-2 to find B. This is extremely efficient and less prone to algebra errors. However, the cover-up rule only works directly for distinct linear factors, not for repeated or quadratic factors.
对于线性因子，一种更快捷的方法是遮盖法：要求出 A，遮住分母中的 (x-1)，将 x=1 代入剩余表达式；同样地，遮住 (x+2) 并代入 x=-2 即可求出 B。这种方法极为高效，且不易出现代数错误。但遮盖法只直接适用于不重复的线性因子，对重复因子或二次因子不适用。
4. Repeated Linear Factors | 重复的线性因子
If the denominator contains a repeated linear factor such as (x-1)², the partial fraction must include ascending powers of the factor: A/(x-1) + B/(x-1)². For (x-1)³, we would include A/(x-1) + B/(x-1)² + C/(x-1)³. The key is to set up the correct form before solving. Neglecting to include all powers is one of the most common errors students make.
如果分母包含重复的线性因子，例如 (x-1)²，部分分式必须包含该因子的升幂形式：A/(x-1) + B/(x-1)²。对于 (x-1)³，需包含 A/(x-1) + B/(x-1)² + C/(x-1)³。关键是先设定正确的分解形式再求解。遗漏某些幂次是学生最常犯的错误之一。
After decomposition, integrating terms like B/(x-1)² is straightforward: it gives -B/(x-1) using the power rule for integration, since (x-1)⁻² integrates to -(x-1)⁻¹. The logarithmic term A ln|x-1| still appears from the first component. Always write the final answer with all terms, and remember that the power rule works for any exponent except -1.
分解后，积分项 B/(x-1)² 非常简单：利用幂函数的积分法则， (x-1)⁻² 的积分为 -(x-1)⁻¹，即 -B/(x-1)。对数项 A ln|x-1| 仍出自第一分式。最终答案应写出所有项，并牢记幂法则适用于除 -1 以外的任何指数。
5. Irreducible Quadratic Factors | 不可约的二次因子
When the denominator has a quadratic factor that cannot be factorised into real linear factors, such as x²+1 or x²+4x+5, the corresponding numerator in the partial fraction must be linear: (Cx+D)/(x²+4x+5). This allows integration using a combination of a logarithm (when the numerator is the derivative of the denominator) and an arctangent (after completing the square).
当分母含有无法分解为实线性因子的二次因子时，如 x²+1 或 x²+4x+5，相应的部分分式分子必须为线性形式：(Cx+D)/(x²+4x+5)。这样就能通过组合对数（当分子是分母的导数时）和反正切（配方后）进行积分。
To integrate (Cx+D)/(x²+4x+5), first check if the numerator can be expressed as a multiple of the derivative 2x+4. Write Cx+D = (C/2)(2x+4) + (D – 2C). The first part integrates to (C/2) ln|x²+4x+5|. The constant remainder is put over (x²+4x+5) and integrated by completing the square: (x+2)²+1, leading to an arctan term.
积分 (Cx+D)/(x²+4x+5) 时，首先检查分子能否表示为导数 2x+4 的倍数。将 Cx+D 写为 (C/2)(2x+4) + (D – 2C)。第一部分积分为 (C/2) ln|x²+4x+5|。常数余项则置于 (x²+4x+5) 中并通过配方积分：(x+2)²+1，从而得到反正切项。
This procedure is particularly common in questions where the denominator cannot be simplified further. Recognising the need for completing the square and the standard integral pattern is essential. Many past Edexcel questions combine partial fractions with such quadratic denominators to test complete mastery of integration techniques.
这种方法在分母无法进一步化简的题目中特别常见。识别出需要配方以及标准积分模式至关重要。许多 Edexcel 历史考题将部分分式与此类二次分母结合，以全面考查积分技巧的掌握程度。
6. Handling Improper Fractions with Division First | 先做长除法处理假分式
A rational function is improper when the degree of the numerator is greater than or equal to the degree of the denominator. Partial fraction decomposition requires a proper fraction, so you must perform polynomial long division before breaking it into partial fractions. The result will be a polynomial plus a proper rational part, which is then decomposed.
当分子次数大于或等于分母次数时，有理函数为假分式。部分分式分解要求真分式，因此必须先进行多项式长除法，再拆分为部分分式。结果将是一个多项式加上一个真有理部分，后者再进行分解。
For example, integrate (x³+2x)/(x²-1). Dividing gives x + (3x)/(x²-1). The remainder (3x)/(x²-1) can now be decomposed using linear factors (x-1)(x+1). Then integrate the polynomial x and the partial fractions separately. Failing to do the division first will lead to an incorrect setup and zero method marks.
例如，积分 (x³+2x)/(x²-1)。通过长除法得到 x + (3x)/(x²-1)。余式 (3x)/(x²-1) 现在可以利用线性因子 (x-1)(x+1) 进行分析。然后分别积分多项式 x 和各个部分分式。若未先做长除法，将导致设定错误，方法分全丢。
7. Step-by-Step Method for Exam Success | 考试成功的分步法
Follow a strict sequence in the exam: (1) Check if the fraction is proper; if not, divide. (2) Factorise the denominator completely. (3) Write the partial fraction form with unknown constants. (4) Multiply through by the original denominator and equate coefficients, or use substitution/cover-up for linear factors. (5) Integrate each term using standard results. (6) Simplify and add the constant of integration.
考试中要严格遵循步骤：(1) 检查分式是否为真分式，若不是，先做长除法。(2) 将分母完全因式分解。(3) 写出含待定常数的部分分式形式。(4) 乘以原分母并比较系数，或对线性因子使用代入/遮盖法。(5) 使用标准结果逐项积分。(6) 化简并加上积分常数。
Presenting your work clearly is vital for gaining method marks even if the final answer contains a sign error. Make sure to state the partial fraction identity explicitly, show the equation used to solve for constants, and then write the integral of each term separately. Edexcel mark schemes reward a clear logical flow.
清晰呈现解题过程对获得方法分至关重要，即使最终答案包含符号错误。要明确写出部分分式恒等式，展示用于求解常数的方程，然后分别写出每一项的积分。Edexcel 评分标准鼓励清晰的逻辑流程。
8. Worked Example 1: Distinct Linear Factors | 示例 1：不重复的线性因子
Find ∫ (5x+3)/[(x-1)(x+2)] dx. First, set (5x+3)/[(x-1)(x+2)] = A/(x-1) + B/(x+2). Multiply by denominator: 5x+3 = A(x+2) + B(x-1). Using the cover-up rule: for A, cover (x-1), let x=1, giving 5(1)+3 = A(3) → A=8/3. For B, cover (x+2), let x=-2, giving 5(-2)+3 = B(-3) → -7 = -3B → B=7/3. Integral = (8/3) ln|x-1| + (7/3) ln|x+2| + C.
求 ∫ (5x+3)/[(x-1)(x+2)] dx。首先设 (5x+3)/[(x-1)(x+2)] = A/(x-1) + B/(x+2)。乘以分母得：5x+3 = A(x+2) + B(x-1)。用遮盖法求 A：遮住 (x-1)，令 x=1，得 5(1)+3 = A(3) → A=8/3。求 B：遮住 (x+2)，令 x=-2，得 5(-2)+3 = B(-3) → -7 = -3B → B=7/3。积分 = (8/3) ln|x-1| + (7/3) ln|x+2| + C。
Always check by combining the fractions back to verify the constants. This takes a moment and prevents careless mistakes. The cover-up method is valid because the factors are linear and distinct. The result can also be written with combined logarithm properties for a slightly neater form, but it is not required.
始终应将分式重新合并回去验证常数，这片刻的检查能防止粗心错误。由于因子为不同线性式，遮盖法有效。结果也可利用对数性质合并成稍简洁的形式，但考试不作要求。
9. Worked Example 2: Repeated Linear Factor | 示例 2：重复的线性因子
Evaluate ∫ (4x+1)/(x-2)² dx. The partial fraction form is A/(x-2) + B/(x-2)². Multiply: 4x+1 = A(x-2) + B. Substitute x=2: 4(2)+1 = B → B=9. Then compare coefficients of x: 4 = A, so A=4. Integral = ∫ [4/(x-2) + 9/(x-2)²] dx = 4 ln|x-2| – 9/(x-2) + C. The minus sign comes from integrating (x-2)⁻²: ∫ (x-2)⁻² dx = – (x-2)⁻¹.
计算 ∫ (4x+1)/(x-2)² dx。部分分式形式为 A/(x-2) + B/(x-2)²。相乘：4x+1 = A(x-2) + B。代入 x=2 得：4(2)+1 = B → B=9。然后比较 x 的系数：4 = A，故 A=4。积分 = ∫ [4/(x-2) + 9/(x-2)²] dx = 4 ln|x-2| – 9/(x-2) + C。负号源自对 (x-2)⁻² 积分：∫ (x-2)⁻² dx = – (x-2)⁻¹。
Notice that the logarithmic term comes from the first partial fraction, while the second gives a rational function. Many students mistakenly apply the logarithm to the B term as well; remember that the integral of 1/(x-2)² is not a logarithm. The power rule applies because the exponent is -2, not -1.
注意，对数项来自第一个部分分式，而第二个给出有理函数。许多学生错误地对 B 项也使用对数；记住，1/(x-2)² 的积分不是对数。应用幂法则因为指数是 -2，而非 -1。
10. Worked Example 3: Quadratic Factor | 示例 3：二次因子
Integrate ∫ (2x²+3x+5)/[(x+1)(x²+2)] dx. Write as A/(x+1) + (Bx+C)/(x²+2). Multiply and solve. Cover-up: A = value at x=-1 = (2(-1)²+3(-1)+5)/( (-1)²+2 ) = (2-3+5)/(1+2) = 4/3. Then expand to find B and C: 2x²+3x+5 = A(x²+2) + (Bx+C)(x+1). Equate coefficients: x²: 2 = A + B → B = 2 – 4/3 = 2/3. Constants: 5 = 2A + C → 5 = 8/3 + C → C = 7/3. Check x coefficient: 3 = B + C = 2/3+7/3=3, consistent. Now integral: (4/3) ln|x+1| + ∫ (2/3 x + 7/3)/(x²+2) dx. Split that integral: (2/3)x/(x²+2) integrates to (1/3) ln(x²+2). The 7/3 part: ∫ 7/3 * 1/(x²+2) dx = (7/3)*(1/√2) arctan(x/√2). Combine and add C.
求 ∫ (2x²+3x+5)/[(x+1)(x²+2)] dx。写为 A/(x+1) + (Bx+C)/(x²+2)。相乘求解。遮盖法：A = x=-1 时的值 = (2(-1)²+3(-1)+5)/( (-1)²+2 ) = (2-3+5)/(1+2) = 4/3。然后展开求 B 和 C：2x²+3x+5 = A(x²+2) + (Bx+C)(x+1)。比较系数：x²: 2 = A + B → B = 2 – 4/3 = 2/3。常数：5 = 2A + C → 5 = 8/3 + C → C = 7/3。检验 x 系数：3 = B + C = 2/3+7/3=3，一致。现在积分：(4/3) ln|x+1| + ∫ (2/3 x + 7/3)/(x²+2) dx。拆分该积分：(2/3)x/(x²+2) 积分为 (1/3) ln(x²+2)。7/3 部分：∫ 7/3 * 1/(x²+2) dx = (7/3)*(1/√2) arctan(x/√2)。合并，并加上 C。
This example demonstrates how three types of integrals emerge from one problem: a logarithm from the linear factor, another logarithm from a derivative numerator, and an arctangent from the constant part. Practice such comprehensive examples to be prepared for the longer 8-10 mark questions in the Edexcel Paper 2 or Paper 3.
此例展示了如何从一个问题中产生三类积分：线性因子带来对数、导数分子带来又一个对数、常数部分带来反正切。多做此类综合练习，以备战 Edexcel 卷二或卷三中 8-10 分的长题。
11. Common Pitfalls and How to Avoid Them | 常见错误与避错指南
Pitfall 1: Forgetting to check the degree of the numerator before starting. Always perform division if the fraction is improper. Pitfall 2: Using the wrong form for repeated or quadratic factors. Stick to the standard forms: for (x-a)ⁿ include all lower powers; for x²+bx+c use Bx+C as the numerator. Pitfall 3: Missing absolute values in log integrals. Unless solving a definite integral with positive limits, always use ln|…|.
错误 1：开始前忘记检查分子次数。若为假分式，一定要先做除法。错误 2：对重复因子或二次因子使用了错误的形式。坚持标准形式：对 (x-a)ⁿ 要包含所有低次幂；对 x²+bx+c 使用 Bx+C 做分子。错误 3：对数积分遗漏绝对值。除非解定积分且区间内表达式恒正，否则始终使用 ln|…|。
Pitfall 4: Mishandling the integration of terms like 1/(x²+a²). The formula is (1/a) arctan(x/a), not arctan(x). The denominator a is the square root of the constant term after completing the square. Pitfall 5: Arithmetic errors when solving for constants. Always substitute a convenient value of x (like a root of the denominator) to find one constant instantly, then equate coefficients systematically.
错误 4：错误处理 1/(x²+a²) 的积分。公式为 (1/a) arctan(x/a)，而非 arctan(x)。分母 a 是配方后常数项的平方根。错误 5：求常数时代数出错。应始终代入方便的 x 值（如分母的根）直接求出某个常数，然后系统地比较系数。
12. Exam Strategy and Final Checks | 考试策略与最终检查
In the exam, partial fractions often appear as a small part of a larger question, such as finding the area enclosed by a curve defined by rational parametric equations or solving a first-order differential equation by separation of variables. Recognising the pattern quickly saves time. Underline the given rational expression and immediately write the decomposition form in the margin before diving into algebra.
考试中，部分分式常作为大题的一小部分出现，比如求由有理参数方程定义的曲线所围面积，或用分离变量法解一阶微分方程。快速识别模式能节省时间。在进入代数运算前，先划出给定的有理式，并立刻在页边写出分解形式。
Once you have integrated, differentiate your result mentally to check if it recovers the original integrand. This mental check takes seconds and confirms the algebraic manipulation. If the question involves a definite integral, apply the limits carefully and check for any negative areas. Finally, present the final answer in simplified exact form, as Edexcel often requires answers in terms of ln and π.
完成积分后，心算微分一下结果，检查是否能回到原被积函数。这种心算只需几秒钟，却能确证代数操作无误。若题目涉及定积分，仔细代入上下限，并检查是否存在负面积。最后，将答案表达为简化精确形式，因为 Edexcel 常要求以 ln 和 π 表示最终结果。
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导