📚 Partial Fractions for Integration | 部分分式积分法
In IB Mathematics, integration often brings us face to face with rational functions that cannot be integrated directly. The method of partial fractions converts a complicated fraction into a sum of simpler ones, each of which can be handled with basic integration rules such as logarithms, inverse tangents or power laws.
在IB数学中，积分经常让我们面对无法直接积出的有理函数。部分分式方法将一个复杂的分式拆成几个简单分式的和，每一个简单分式都可以用对数、反正切或幂函数等基本积分规则来处理。
1. Introduction to Partial Fractions | 部分分式简介
x where both f and g are polynomials, the key idea is to express the integrand as a sum of 'partial fractions' whose denominators are factors of g(x). Once decomposed, the integral reduces to a handful of standard forms.
当我们试图计算 ∫ f(x)/g(x) dx，其中 f 和 g 都是多项式时，关键思想是将被积函数表示为一组“部分分式”的和，这些部分分式的分母恰好是 g(x) 的因式。一旦拆开，积分就归结为若干标准形式。
The technique is particularly powerful when g(x) can be factorised completely into linear and irreducible quadratic factors over the real numbers.
当 g(x) 在实数范围内可以完全分解为一次因式和不可约二次因式时，这一技巧格外有力。
2. Conditions for Decomposition: Proper and Improper Fractions | 分解条件：真分式与假分式
A rational expression f(x)/g(x) is called proper if the degree of the numerator is strictly less than the degree of the denominator. If it is not proper (i.e. improper), we must first perform polynomial division to obtain a polynomial plus a proper rational part.
若分子多项式的次数严格低于分母的次数，则有理式 f(x)/g(x) 称为真分式。如果不是真分式（即假分式），我们必须先做多项式除法，得到一个多项式加上一个真有理部分。
Partial fraction decomposition only applies directly to the proper fraction. Any improper fraction should be rewritten as Q(x) + R(x)/g(x), where R(x) is of lower degree than g(x). Then we decompose R(x)/g(x).
部分分式分解仅直接适用于真分式。任何假分式都应改写为 Q(x) + R(x)/g(x) 的形式，其中 R(x) 的次数低于 g(x)。然后再对 R(x)/g(x) 进行分解。
3. Distinct Linear Factors | 不同的一次因式
Suppose the denominator factorises into distinct linear factors: (a₁x + b₁)(a₂x + b₂)...(aₙx + bₙ). Then the proper rational expression can be written as a sum of terms Aᵢ/(aᵢx + bᵢ), where the constants Aᵢ are determined.
假设分母分解为不同的一次因式：(a₁x + b₁)(a₂x + b₂)...(aₙx + bₙ)。那么该真有理式可以写为若干项 Aᵢ/(aᵢx + bᵢ) 的和，其中常数 Aᵢ 待定。
+ B/(x-2) and solve for A and B. Each term integrates to a logarithm: ∫ A/(ax+b) dx = (A/|a|) ln|ax+b| + C.
例如，如果分母是 (x+1)(x-2)，我们把分式写成 A/(x+1) + B/(x-2) 并解出 A 和 B。每一项积分得到对数函数：∫ A/(ax+b) dx = (A/|a|) ln|ax+b| + C。
| Denominator factor | Partial fraction term |
| ax + b | A/(ax + b) |
| (ax + b)(cx + d) | A/(ax + b) + B/(cx + d) |
4. Repeated Linear Factors | 重复的一次因式
If the denominator contains a repeated linear factor (ax + b)ⁿ, the partial fraction decomposition must include a chain of terms: A₁/(ax + b) + A₂/(ax + b)² + ... + Aₙ/(ax + b)ⁿ.
如果分母含有重复的一次因式 (ax + b)ⁿ，部分分式分解必须包含一系列项：A₁/(ax + b) + A₂/(ax + b)² + ... + Aₙ/(ax + b)ⁿ。
This is because a single term would not be sufficient to represent the original fraction. For example, with (x-2)² in the denominator, we write the decomposition as B/(x-2) + C/(x-2)², not just C/(x-2)².
这是因为仅有一项不足以表示原分式。例如，分母有 (x-2)²，我们分解为 B/(x-2) + C/(x-2)²，而不是仅有 C/(x-2)²。
power of (ax+b) in the denominator: ∫ A/(ax+b)ⁿ dx = -A/[(n-1)a(ax+b)ⁿ⁻¹] + C, for n > 1.
积分时，指数 n > 1 的项在分母中出现 (ax+b) 的幂：∫ A/(ax+b)ⁿ dx = -A/[(n-1)a(ax+b)ⁿ⁻¹] + C（n > 1）。
5. Irreducible Quadratic Factors | 不可约二次因式
An irreducible quadratic factor is one that cannot be written as a product of real linear factors, for instance x² + 1 or x² + x + 1 (with discriminant Δ < 0). In the decomposition, such a factor ax² + bx + c contributes a term of the form (Ax + B)/(ax² + bx + c).
不可约二次因式是指不能写成实系数一次因式乘积的二次式，例如 x² + 1 或 x² + x + 1（判别式 Δ < 0）。分解时，这样的因式 ax² + bx + c 带来一项形如 (Ax + B)/(ax² + bx + c)。
Integration of (Ax + B)/(ax² + bx + c) usually splits into two parts: one yields a natural logarithm via the derivative of the denominator, and the other yields an arctangent after completing the square.
(Ax + B)/(ax² + bx + c) 的积分通常拆成两部分：一部分通过分母的导数给出自然对数，另一部分通过配方法给出反正切。
an be handled by writing numerator as derivative of denominator plus a constant, leading to ln|x²+2x+5| + arctan term.
例如，∫ (2x+3)/(x²+2x+5) dx 可将分子写成分母的导数加常数，从而得到 ln|x²+2x+5| 和反正切项。
6. Repeated Irreducible Quadratic Factors | 重复的不可约二次因式
When an irreducible quadratic factor appears with multiplicity m, say (ax² + bx + c)ᵐ, the decomposition requires terms of the form (A₁x + B₁)/(ax² + bx + c) + (A₂x + B₂)/(ax² + bx + c)² + ... + (Aₘx + Bₘ)/(ax² + bx + c)ᵐ.
当不可约二次因式以重数 m 出现时，即 (ax² + bx + c)ᵐ，分解需要形如 (A₁x + B₁)/(ax² + bx + c) + (A₂x + B₂)/(ax² + bx + c)² + ... + (Aₘx + Bₘ)/(ax² + bx + c)ᵐ 的项。
Integrating the higher powers of a quadratic denominator may require reduction formulas or trigonometric substitution, but in IB questions such repetitions are usually kept simple or reduced by other techniques.
积分高次二次分母的幂可能需要递推公式或三角换元，不过在IB题目中这类重复通常保持简单，或通过其他技巧化简。
7. Handling Improper Fractions: Polynomial Division | 处理假分式：多项式长除法
at f(x)/g(x) = Q(x) + R(x)/g(x). The partial fractions are then applied only to R(x)/g(x).
若分子次数 ≥ 分母次数，我们必须先用长除法或多项式除法。得到商式 Q(x) 和余式 R(x)，于是 f(x)/g(x) = Q(x) + R(x)/g(x)。然后只对 R(x)/g(x) 使用部分分式。
+1) - x, so x³/(x²+1) = x - x/(x²+1). Then ∫ x dx - ∫ x/(x²+1) dx = ½ x² - ½ ln(x²+1) + C.
示例：∫ x³/(x²+1) dx。除法给出 x³ = x(x²+1) - x，所以 x³/(x²+1) = x - x/(x²+1)。然后 ∫ x dx - ∫ x/(x²+1) dx = ½ x² - ½ ln(x²+1) + C。
8. Solving for Coefficients: Methods | 求解系数的方法
Once we set up the partial fraction form with unknown constants, we need efficient methods to find them. The three standard approaches are substitution of convenient x-values (the cover‑up method), equating coefficients, and using differentiation or limits.
一旦搭建好含有未知常数的部分分式形式，我们需要高效地求出它们。三种标准方法是：代入方便的 x 值（遮盖法）、比较系数法以及使用求导或极限的方法。
Cover‑ o the remaining expression. This works only for denominators with distinct linear factors.
遮盖法：对于不同的一次因式，要找到因式 (x - a) 对应的系数 A，把原分母中的 (x - a) 遮住，然后代入 x = a 到剩余表达式中。这仅适用于分母为不同一次因式的情况。
Equating coefficients: Multiply both sides by the full denominator, expand, and match coefficients of like powers of x on both sides. This is general and reliable for any decomposition.
比较系数法：将等式两边同乘以整个分母，展开后比较两边同次幂的系数。这是一种通用且可靠的方法，适用于所有分解。
s A = ½. Cover (x+2) and set x=-2 gives B = -½. This is faster than equating coefficients.
示例：分解 1/(x(x+2)) = A/x + B/(x+2)，遮住 x 令 x=0 得 A = ½。遮住 (x+2) 令 x=-2 得 B = -½。这比比较系数法更快。
9. Integration of Each Partial Fraction Term | 各部分分式的积分
After decomposition, we integrate term by term. The elementary integration results required are:
分解之后，我们逐项积分。所需的基本积分公式为：
∫ A/(ax + b) dx = (A/|a|) ln|ax + b| + C.
∫ A/(ax + b)ⁿ dx = -A/[(n-1)a(ax + b)ⁿ⁻¹] + C, n > 1.
dx = (A/2) ln(x² + c²) + (B/c) arctan(x/c) + C (after completing the square if necessary).
∫ A/(ax + b) dx = (A/|a|) ln|ax + b| + C。
∫ A/(ax + b)ⁿ dx = -A/[(n-1)a(ax + b)ⁿ⁻¹] + C（n > 1）。
∫ (Ax + B)/(x² + c²) dx = (A/2) ln(x² + c²) + (B/c) arctan(x/c) + C（必要时先配方）。
For more complicated irreducible quadratics, we split the numerator into the derivative of the denominator plus a constant, then integrate the logarithmic part and the arctangent part separately.
对于更复杂的不可约二次式，我们将分子拆成分母的导数加常数，然后分别求出对数部分和反正切部分的积分。
10. Worked Examples and Common Mistakes | 例题和常见错误
Example 1: Evaluate ∫ (3x+5)/((x+1)(x+2)) dx.
Write (3x+5)/((x+1)(x+2)) = A/(x+1) + B/(x+2).
Cover (x+1) and set x=-1: A = (3(-1)+5)/(-1+2) = 2/1 = 2.
Cover (x+2) and set x=-2: B = (3(-2)+5)/(-2+1) = (-1)/(-1) = 1.
Hence ∫ [2/(x+1) + 1/(x+2)] dx = 2 ln|x+1| + ln|x+2| + C.
例题1：计算 ∫ (3x+5)/((x+1)(x+2)) dx。
写 (3x+5)/((x+1)(x+2)) = A/(x+1) + B/(x+2)。
遮盖 (x+1) 代入 x=-1：A = (3(-1)+5)/(-1+2) = 2/1 = 2。
遮盖 (x+2) 代入 x=-2：B = (3(-2)+5)/(-2+1) = (-1)/(-1) = 1。
于是 ∫ [2/(x+1) + 1/(x+2)] dx = 2 ln|x+1| + ln|x+2| + C。
Example 2 (repeated linear): ∫ (x)/(x-1)² dx. Since fraction is improper? ompare coefficients: A = 1, B - A = 0 → B = 1. Integral: ∫ 1/(x-1) dx + ∫ 1/(x-1)² dx = ln|x-1| - 1/(x-1) + C.
例题2（重复一次因式）：∫ (x)/(x-1)² dx。分子次数等于分母，但可分解：x/(x-1)² = A/(x-1) + B/(x-1)²。通分：x = A(x-1) + B → x = Ax + (B - A)。比较系数：A = 1, B - A = 0 → B = 1。积分：∫ 1/(x-1) dx + ∫ 1/(x-1)² dx = ln|x-1| - 1/(x-1) + C。
Common mistakes: forgetting to check for improper fraction; missing a term in repeated factors; not splitting quadratic numerator properly; using cover‑up method on repeated or
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