📚 pdfjoiner_(4)-176: Partial Fractions & Integration | pdfjoiner_(4)-176：部分分式与积分
fraction into simpler components not only deepens your algebraic fluency but also unlocks powerful integration techniques that are regularly examined in Edexcel papers. This article – inspired by the pdfjoiner_(4)-176 problem set – de range of integrals. Whether you are preparing for your mock or the final examination, mastering this topic will give you confidence and extra marks across Pure Mathematics 3 and beyond.
将有理式拆分成部分分式是A-Level纯数的重要基石。把一个复杂的单一分式拆成几个简单部分的能力，不仅能加深你的代数运算流畅度，还能解锁强有力的积分技巧——这正是Edexcel试卷中频繁考查的内容。本文以pdfjoiner_(4)-176习题集为蓝本，系统覆盖了你可能遇到的所有部分分式类型、如何对它们进行分解，以及如何利用这些分解结果计算各类积分。无论你正在准备模拟考还是最终大考，掌握这一主题都会让你在Pure Mathematics 3乃至更广的范围里充满信心、多拿分数。
1. What Are Partial Fractions? | 什么是部分分式？
A partial fraction is simply a simpler algebraic fraction that, when added to others, reconstructs a given rational function. If we are given an expression such as (3x+5)/(x²+x-2), we can break it into a sum of fractions like A/(x+2) + B/(x-1). This decomposition is possible provided the degree of the numerator is strictly less than the degree of the denominator – in other words, the fraction must be proper. The constants A and B are found by equating coefficients or substituting convenient values of x. Once you have these basic ideas, you can handle everything from simple linear factors to irreducible quadratics.
部分分式是指一个更简单的代数分式，当与其他部分分式相加时，能够重构给定的有理函数。例如给出表达式(3x+5)/(x²+x-2)，我们可以将其拆分为A/(x+2) + B/(x-1)的和。只要分子的次数严格小于分母的次数——换句话说，该分式必须是真分式——这种分解就可以实现。常数A和B可以通过比较系数或代入便利的x值来确定。一旦掌握了这些基本思路，你就能应对从简单线性因子到不可约二次式的各种情况。
2. Proper vs Improper Fractions | 真分式与假分式
Before attempting any partial fraction decomposition, always check the degrees of the numerator and denominator. If the numerator has equal or higher degree – for example (x³+2)/(x²-1) – the fraction is improper. In such cases you must first perform polynomial division to rewrite it as a polynomial plus a proper fraction. Only the proper fraction part is then decomposed. In Edexcel exams, the need for division is often tested explicitly, so never skip this step.
在尝试任何部分分式分解之前，请务必先检查分子和分母的次数。如果分子的次数等于或高于分母——比如(x³+2)/(x²-1)——这个分式就是假分式。此时你必须首先进行多项式除法，将其改写为一个多项式加上一个真分式。随后只对真分式部分进行分解。在Edexcel考试中，需要做除法的考点经常被直接考查，因此千万不要跳过这一步。
3. Linear Factors: The Foundation Case | 线性因子：基础情况
ltiply through by the common denominator and equate coefficients, or substitute x = -2 and x = 1 to find A and B directly. This is the most straightforward case and is the first type you will learn. Always remember to check your constants by combining the fractions again to see if you get the original expression.
当分母可以分解为不同的线性因子时，例如(x+2)(x-1)，我们设分解形式为A/(x+2) + B/(x-1)。两边同乘公分母，再比较系数，或者直接代入x = -2和x = 1来求出A和B。这是最简单直接的情况，也是你会最先学到的类型。一定要记住通过将分式重新合并来检验常数是否正确，看看能否还原原先的表达式。
4. Repeated Linear Factors | 重复线性因子
If the denominator contains a factor like (x-1)², the decomposition must include terms for all powers of that factor down to the first power. For a squared factor we write A/(x-1)² + B/(x-1). For a cubed factor, we use A/(x-1)³ + B/(x-1)² + C/(x-1). Failing to include the lower powers is one of the most common mistakes in A-Level exams. Always count the multiplicity and include one term for each descending power.
如果分母含有如(x-1)²这样的因子，分解时必须包含该因子的所有幂次项，从最高次一直降到一次。对于平方因子，我们写成A/(x-1)² + B/(x-1)。对于立方因子，则使用A/(x-1)³ + B/(x-1)² + C/(x-1)。漏写较低次幂项是A-Level考试中最常见的错误之一。一定要根据重数计数，为每个递减的幂次都写出一项。
5. Irreducible Quadratic Factors | 不可约二次因子
Some denominators contain quadratic expressions that cannot factorise over the real numbers, such as x²+1 or x²+2x+5. In these cases the corresponding partial fraction numerator needs to be linear: (Ax + B)/(x²+1). Learning to work with such factors is crucial because they often lead to integrals involving the arctan function. The constants A and B are still found by multiplying out and comparing coefficients of x³, x², x and the constant term.
有些分母含有不能在实数范围内分解的二次式，例如x²+1或x²+2x+5。在这些情况下，对应的部分分式的分子需要是线性式：(Ax + B)/(x²+1)。学会处理这类因子至关重要，因为它们常会引出涉及arctan函数的积分。常数A和B的求法仍然是通过乘开并比较x³项、x²项、x项和常数项的系数来获得。
6. The General Strategy for Decomposition | 分解的通用策略
A systematic approach avoids careless errors. First, check if the fraction is proper; if not, divide. Second, fully factorise the denominator into linear and irreducible quadratic factors. Third, write the general form with unknown constants. Fourth, multiply through by the denominator and either equate coefficients or substitute suitable x-values. Finally, solve the resulting system of linear equations. With practice, this sequence becomes second nature and you can complete most exam decomposition questions in under four minutes.
有系统的处理方式能避免粗心错误。第一步，检查分式是否为真分式；若不是，先做除法。第二步，将分母完全分解为线性因子和不可约二次因子。第三步，写出含未知常数的通用形式。第四步，乘以分母，然后比较系数或代入合适的x值。最后，解所得线性方程组。通过练习，这一流程会变成你的第二天性，你可以在四分钟内完成绝大多数考试中的分解题目。
7. Partial Fractions and Integration – The Link | 部分分式与积分——关键联系
The reason partial fractions are emphasised so strongly in Edexcel A-Level mathematics is their direct application to integration. Many rational functions that look impossible to integrate directly become trivial once decomposed. Terms of the form A/(ax+b) integrate to (A/a) ln|ax+b|, while terms like (Ax+B)/(ax²+bx+c) often split into a logarithmic part and an arctan part after completing the square. Recognising this connection is half the battle in topic 8 of the Pure Mathematics specification.
Edexcel A-Level数学中如此强调部分分式的原因，是因为它们直接应用于积分。许多看似无法直接积分的有理函数，一旦被分解就会变得极其简单。A/(ax+b)形式的项可以积分为(A/a) ln|ax+b|，而像(Ax+B)/(ax²+bx+c)这样的项在配方后通常可以拆分成一个对数部分和一个arctan部分。认识到这种联系，就已经解决了纯数考试大纲第8章中一半的难题。
8. Integrating Linear-Factor Partial Fractions | 积分线性因子的部分分式
2x-3) dx. The denominator factorises as (x-3)(x+1). Decomposition gives 3/(x-3) + 2/(x+1). The integral then becomes 3 ln|x-3| + 2 ln|x+1| tegration, and be careful with the modulus signs inside the logarithm. This is a classic exam question that can appear in both pure and applied contexts, particularly when evaluating definite integrals between given limits.
考虑积分 ∫ (5x-3)/(x²-2x-3) dx。分母分解为(x-3)(x+1)。分解后得到3/(x-3) + 2/(x+1)。于是该积分变为 3 ln|x-3| + 2 ln|x+1| + C。一定要写上积分常数，并注意对数里的绝对值符号。这是一道经典的考试题，既可以出现在纯数部分，也可以出现在应用情境中，尤其是在计算给定上下限的定积分时。
9. Handling Repeated Linear Factors in Integrals | 处理积分中的重复线性因子
nto A/(x-2)² + B/(x-2), you obtain integrals like ∫ A/(x-2)² dx = -A/(x-2) and ∫ B/(x-2) dx = B ln|x-2|. The power rule used for (x-2)⁻² is a common source of sign errors — u⁻² du = -u⁻¹. Practice this with the pdfjoiner_(4)-176 set, especially problems 32 to 40.
当分母含有重复因子时，积分得到的各项就需要特别小心。例如，把(4x+1)/(x-2)²分解成A/(x-2)² + B/(x-2)后，你会得到像∫ A/(x-2)² dx = -A/(x-2)和 ∫ B/(x-2) dx = B ln|x-2|这样的积分。对(x-2)⁻²运用幂函数法则经常出现符号错误——记住∫ u⁻² du = -u⁻¹。请用pdfjoiner_(4)-176题集中的题目进行练习，特别是第32题到第40题。
10. Definite Integrals and Substitution | 定积分与代入法
partial fractions. For instance, in ∫₀¹ (x+1)/(x²+4x+3) dx, factorise the denominator as (x+1)(x+3), cancel the common factor, then integrate the simplified proper fraction. Cancellation is a trick that simplifies work immensely, but you must state the domain restrictions clearly. Always check whether a factor is zero within the integration interval; if so, the integral may be improper.
当给定积分带有上下限时，既可以积分后代回原变量，也可以在结合部分分式使用换元法时改变积分限。例如，在积分∫₀¹ (x+1)/(x²+4x+3) dx中，将分母分解为(x+1)(x+3)，约去公因子，然后对简化后的真分式进行积分。约分这一技巧能极大简化计算，但你必须清楚地说明自变量的限制。始终要检查在积分区间内是否有因子为零；如果有，该积分可能是一个反常积分。
11. Quadratic Denominators and the Arctan Form | 二次分母与Arctan形式
. However, completing the square gives (x+1)²+4. The form ∫ 1/(u²+a²) du = (1/a) arctan(u/a) then applies directly. Sometimes a partial fraction decomposition yields a linear numerator over a quadratic denominator; separating into a multiple of the derivative of the denominator plus a constant leaves a ln term and an arctan term. This technique appears almost every year on Edexcel P3 papers.
像∫ 1/(x²+2x+5) dx这样的积分，无法仅凭实分母的简单部分分式来解决，因为该二次式是不可约的。然而，配方后得到(x+1)²+4。然后可以直接套用∫ 1/(u²+a²) du = (1/a) arctan(u/a)的形式。有时候，部分分式分解会得出分子为线性、分母为二次式的形式；将其拆分为分母导数的倍数加上一个常数，就可以分离出一个ln项和一个arctan项。这一技巧几乎每年都会出现在Edexcel P3试卷上。
12. Mastery Through Practice and the 176 Problem Set | 通过练习与176题集达到精通
ins exactly 176 carefully chosen problems that cover every nuance of partial fractions and their integration. Work through them systematically: start with simple proper fractions, progress to repeated and quadratic factors, and finish with challenging mixed integrals. For each problem, write down the decomposition form first, find constants, perform the integration, and then differentiate your answer to check. Building this discipline not only secures high marks in algebraic methods and integration but also prepares you for the kind of multi-step problem solving that distinguishes top Edexcel candidates.
pdfjoiner_(4)-176系列恰好包含176道精心挑选的习题，覆盖了部分分式及其积分的每一个细微之处。请系统地进行练习：从简单的真分式开始，逐步过渡到重复因子和二次因子，最后完成具有挑战性的混合积分。每道题目都先写出分解形式，求出常数，完成积分，然后对你的答案求导进行检验。养成这种严谨的习惯，不仅能确保你在代数方法和积分部分拿到高分，还能让你为那种多步骤解决问题的考题做好充分准备——这正是顶尖Edexcel考生脱颖而出的关键。
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