📚 Integration using Partial Fractions | 利用部分分式积分
In A-Level Mathematics, integration of rational functions often requires expressing a complex fraction as a sum of simpler fractions, known as partial fractions. This technique is essential for integrating expressions that cannot be simplified by basic methods, and it appears regularly on Edexcel Pure Mathematics papers.
在A-Level数学中，有理函数的积分通常需要将复杂分式表示为简单分式之和，即部分分式。对于无法用基本方法简化的表达式，这种技巧至关重要，并且在Edexcel纯数试卷中经常出现。
1. Understanding Rational Functions | 理解有理函数
x). When the degree of the numerator is less than the degree of the denominator, it is called a proper rational function; otherwise, it is improper and must be simplified by polynomial division first.
有理函数是任何可以表示为两个多项式之比的函数，例如 f(x) = P(x)/Q(x)。当分子次数低于分母次数时，称为真分式；否则为假分式，并且必须先通过多项式除法进行化简。
2. Factorising the Denominator | 分解分母
The first step in partial fraction decomposition is to factorise the denominator Q(x) into linear factors (e.g. x - a), repeated linear factors (e.g. (x - a)²), or irreducible quadratic factors (e.g. x² + bx + c where the discriminant is negative). A correct factorisation is vital for the success of the method.
部分分式分解的第一步是将分母 Q(x) 分解为线性因式（如 x - a）、重复线性因式（如 (x - a)²）或不可约二次因式（如 x² + bx + c，其判别式为负）。正确因式分解对该方法的成功至关重要。
3. Linear Factors Decomposition | 线性因式分解
he proper fraction can be written as A/(x - a) + B/(x - b). The constants A and B are found by multiplying through by the denominator and either equating coefficients or substituting suitable x-values.
如果分母包含不同的一次因式，例如 Q(x) = (x - a)(x - b)，真分式可写成 A/(x - a) + B/(x - b)。常数 A 和 B 的求解方法有：两边乘以分母后比较系数，或代入合适的 x 值。
Example: 1/((x - 2)(x + 3)) = A/(x - 2) + B/(x + 3)
ply both sides by (x - 2)(x + 3): 1 = A(x + 3) + B(x - 2). Substituting x = 2 gives 1 = 5A ⇒ A = 1/5. Substituting x = -3 gives 1 = -5B ⇒ B = -1/5. So the partial fractions are 1/(5(x - 2)) - 1/(5(x + 3)).
两边乘以 (x - 2)(x + 3)：1 = A(x + 3) + B(x - 2)。代入 x = 2 得 1 = 5A ⇒ A = 1/5。代入 x = -3 得 1 = -5B ⇒ B = -1/5。因此部分分式为 1/(5(x - 2)) - 1/(5(x + 3))。
4. Integration of Linear Factor Terms | 线性因式项的积分
Each partial fraction of the form A/(x - a) integrates to A ln|x - a| + C. For the above example, ∫ [1/(5(x - 2)) - 1/(5(x + 3))] dx = (1/5) ln|x - 2| - (1/5) ln|x + 3| + C = (1/5) ln| (x - 2)/(x + 3) | + C.
每个 A/(x - a) 形式的部分分式积分为 A ln|x - a| + C。对于上例，∫ [1/(5(x - 2)) - 1/(5(x + 3))] dx = (1/5) ln|x - 2| - (1/5) ln|x + 3| + C = (1/5) ln| (x - 2)/(x + 3) | + C。
5. Repeated Linear Factors | 重复线性因式
ance, 1/((x - 1)²(x + 2)) expands as A/(x - 1) + B/(x - 1)² + C/(x + 2). The constant B can then be integrated using the power rule: ∫ B/(x - 1)² dx = -B/(x - 1) + C.
当分母包含因子 (x - a)ⁿ 时，部分分式必须包括分母为 (x - a)、(x - a)²、直至 (x - a)ⁿ 的项。例如，1/((x - 1)²(x + 2)) 可分解为 A/(x - 1) + B/(x - 1)² + C/(x + 2)。常数 B 对应的项可利用幂法则积分：∫ B/(x - 1)² dx = -B/(x - 1) + C。
6. Irreducible Quadratic Factors | 不可约二次因式
For an irreducible quadratic factor like x² + bx + c, the partial fraction term is (Ax + B)/(x² + bx + c). To integrate such a term, we often complete the square in the denominator and use substitution to split the numerator, leading to logarithmic and arctan integrals.
对于 x² + bx + c 等不可约二次因式，部分分式项为 (Ax + B)/(x² + bx + c)。要积分这类项，我们常将分母配成完全平方，并用代换法拆分分子，从而得到对数积分和反正切积分。
7. Handling Improper Fractions | 处理假分式
If the degree of the numerator is greater than or equal to that of the denominator, polynomial long division must be performed first to obtain a polynomial plus a proper fraction. The proper fraction is then decomposed into partial fractions, while the polynomial part can be integrated term by term.
如果分子的次数大于或等于分母的次数，必须先进行多项式长除法，得到一个多项式加一个真分式。然后再将真分式分解为部分分式，而多项式部分可以逐项积分。
8. Integration Strategy with Partial Fractions | 使用部分分式的积分策略
n form with unknown constants; (4) Solve for the constants using any suitable method; (5) Integrate each term separately using standard results like ∫ 1/(x - a) dx = ln|x - a| + C and ∫ 1/(x² + a²) dx = (1/a) arctan(x/a) + C.
要积分有理函数：(1) 若是假分式先做长除法；(2) 将分母完全分解因式；(3) 写出带有未知常数的部分分式形式；(4) 选用恰当方法求出常数；(5) 使用标准结果分别积分每一项，如 ∫ 1/(x - a) dx = ln|x - a| + C 和 ∫ 1/(x² + a²) dx = (1/a) arctan(x/a) + C。
9. Common Pitfalls and Tips | 常见误区与提示
Always check the degree of numerator and denominator first; many students forget the division step. The cover-up method works only for distinct linear factors. For repeated or quadratic factors, equating coefficients is more reliable. Also, remember to include the constant of integration in the final answer.
务必先检查分子分母次数；许多学生常忘记除法步骤。遮盖法仅适用于不同的线性因式。对于重复因式或二次因式，比较系数法更为可靠。同时，记住最终答案中要加上积分常数。
10. Worked Example with Quadratic Factor | 带二次因式的例题
² + 1) + (Bx + C)(x + 1). Comparing coefficients yields A = 1, B = -1, C = 1. The integral becomes ∫ 1/(x + 1) dx + ∫ (-x + 1)/(x² + 1) dx. Split the second integral: -∫ x/(x² + 1) dx + ∫ 1/(x² + 1) dx = -(1/2) ln(x² + 1) + arctan x. Final answer: ln|x + 1| - (1/2) ln(x² + 1) + arctan x + C.
求 ∫ (2x + 1)/((x + 1)(x² + 1)) dx。首先设立部分分式：(2x + 1)/((x + 1)(x² + 1)) = A/(x + 1) + (Bx + C)/(x² + 1)。两边乘以分母：2x + 1 = A(x² + 1) + (Bx + C)(x + 1)。比较系数得 A = 1, B = -1, C = 1。积分变为 ∫ 1/(x + 1) dx + ∫ (-x + 1)/(x² + 1) dx。拆分第二个积分：-∫ x/(x² + 1) dx + ∫ 1/(x² + 1) dx = -(1/2) ln(x² + 1) + arctan x。最终答案：ln|x + 1| - (1/2) ln(x² + 1) + arctan x + C。
11. Table of Common Partial Fraction Forms | 常见部分分式形式表
Below is a summary table for quick reference when setting up the decomposition.
以下是一个总结表格，方便在设立分解时快速查阅。
| Denominator Factors | Partial Fraction Form |
|---|---|
| Distinct linear: (x - a)(x - b) | A/(x - a) + B/(x - b) |
| Repeated linear: (x - a)² | A/(x - a) + B/(x - a)² |
| Irreducible quadratic: (x² + bx + c) | (Ax + B)/(x² + bx + c) |
| Mixed: (x - a)(x² + bx + c) | A/(x - a) + (Bx + C)/(x² + bx + c) |
12. Applications in Real Problems | 实际问题中的应用
Partial fractions are not merely an algebraic exercise; they are crucial in solving differential equations, evaluating Laplace transforms, and in physics and engineering where rational functions model rates of change, such as in chemical reaction kinetics and electric circuits.
部分分式并非单纯的代数练习；它在解微分方程、计算拉普拉斯变换以及在物理与工程中均有重要应用。例如，在化学反应速率和电路分析中，常用有理函数建模变化率，此时部分分式不可或缺。
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