📚 Mathematical Techniques: Core Skills and Methods | 数学：Mathematical Techniques 知识点精讲
Mathematics at A-Level requires a solid grasp of various techniques. This article revisits essential methods across algebra, calculus, trigonometry, and other key areas, providing clear explanations in both English and Chinese.
A-Level数学要求学生扎实掌握各种技巧。本文重温代数、微积分、三角学及其他核心领域的基本方法，并提供清晰的中英双语解析。
1. Algebraic Manipulation | 代数运算技巧
Algebraic manipulation is foundational. Always remember to expand brackets correctly using the distributive law: a(b + c) = ab + ac. When factorising, look for common factors, difference of squares a² – b² = (a – b)(a + b), and perfect squares.
代数运算是基础。牢记正确使用分配律去括号：a(b + c) = ab + ac。因式分解时，寻找公因式、平方差 a² – b² = (a – b)(a + b) 以及完全平方。
When simplifying rational expressions, factor numerator and denominator and cancel common factors, but be mindful of domain restrictions (denominator ≠ 0).
在化简有理式时，分解分子和分母并约去公因式，但要注意定义域限制（分母 ≠ 0）。
Completing the square is a crucial technique for quadratic equations: x² + bx = (x + b/2)² – (b/2)².
配方法是二次方程的重要技巧：x² + bx = (x + b/2)² – (b/2)²。
2. Partial Fractions | 部分分式分解
To split a rational function into partial fractions, first factor the denominator. For distinct linear factors (ax + b)(cx + d), assume the form A/(ax+b) + B/(cx+d). Solve for A and B by equating coefficients or substituting convenient values of x.
为将有理函数分解为部分分式，先分解分母。对于不同的一次因式 (ax + b)(cx + d)，假设形式 A/(ax+b) + B/(cx+d)，通过比较系数或代入特殊 x 值解出 A 和 B。
For repeated linear factors (ax+b)ⁿ, the decomposition includes terms A₁/(ax+b) + A₂/(ax+b)² + … + Aₙ/(ax+b)ⁿ.
对于重复一次因式 (ax+b)ⁿ，分解式包含项 A₁/(ax+b) + A₂/(ax+b)² + … + Aₙ/(ax+b)ⁿ。
For irreducible quadratic factors (ax² + bx + c), include a term (Px + Q)/(ax² + bx + c). Partial fractions are very useful in integration.
对于不可约二次因式 (ax² + bx + c)，包括一项 (Px + Q)/(ax² + bx + c)。部分分式在积分中非常有用。
3. Binomial Expansion | 二项式展开
The binomial theorem for (1 + x)ⁿ, where n is a real number and |x| < 1, is (1 + x)ⁿ = 1 + nx + n(n–1)/2! x² + n(n–1)(n–2)/3! x³ + ... The expansion is infinite and convergent only for |x| < 1.
二项式定理，对 (1 + x)ⁿ，n 为实数且 |x| < 1，展开为 1 + nx + n(n–1)/2! x² + n(n–1)(n–2)/3! x³ + ... 该展开是无穷的，且只有当 |x| < 1 时收敛。
For positive integer n, the expansion is finite and uses binomial coefficients: (a + b)ⁿ = Σₖ₌₀ⁿ C(n, k) aⁿ⁻ᵏ bᵏ, where C(n, k) = n!/(k!(n–k)!).
对于正整数 n，展开是有限的，利用二项式系数：(a + b)ⁿ = Σₖ₌₀ⁿ C(n, k) aⁿ⁻ᵏ bᵏ，其中 C(n, k) = n!/(k!(n–k)!)。
Always check the range of validity, e.g., for (a + bx)ⁿ, factor out aⁿ to get the form aⁿ(1 + (b/a)x)ⁿ.
务必检查有效范围，例如对于 (a + bx)ⁿ，提取 aⁿ 化为 aⁿ(1 + (b/a)x)ⁿ 的形式。
4. Exponential and Logarithmic Functions | 指数与对数函数
Key properties: eˣ and ln x are inverse functions. ln(eˣ) = x, e^(ln x) = x. Product rule: ln(ab) = ln a + ln b. Quotient rule: ln(a/b) = ln a – ln b. Power rule: ln(aⁿ) = n ln a. Note that ln x is defined only for x > 0.
关键性质：eˣ 和 ln x 互为反函数。ln(eˣ) = x，e^(ln x) = x。积的法则：ln(ab) = ln a + ln b。商的法则：ln(a/b) = ln a – ln b。幂的法则：ln(aⁿ) = n ln a。注意 ln x 仅在 x > 0 时有定义。
Derivative: d/dx eˣ = eˣ, d/dx aˣ = aˣ ln a. Integral: ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C. To differentiate xˣ, rewrite using e^(x ln x).
导数：d/dx eˣ = eˣ，d/dx aˣ = aˣ ln a。积分：∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C。要对 xˣ 求导，可改写为 e^(x ln x) 再求导。
5. Trigonometric Identities and Equations | 三角恒等式与方程
Fundamental identities: sin²θ + cos²θ = 1, tan θ = sin θ / cos θ. Double-angle: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ.
基本恒等式：sin²θ + cos²θ = 1，tan θ = sin θ / cos θ。倍角：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。
When solving trigonometric equations, use identities to simplify to a single trig function. Then find principal solutions and add period (2π for sin/cos, π for tan). Always check the domain; additional solutions may arise from squaring.
解三角方程时，运用恒等式化简为单一三角函数的方程。求出主解后加上周期（sin/cos 为 2π，tan 为 π）。务必检查定义域；平方可能导致增根。
For equations like a sin θ + b cos θ = c, write in the form R sin(θ + α) or R cos(θ – α) using R = √(a² + b²) and tan α = b/a.
对于 a sin θ + b cos θ = c 型方程，可化为 R sin(θ + α) 或 R cos(θ – α) 形式，其中 R = √(a² + b²)，tan α = b/a。
6. Differentiation Techniques | 微分技巧
Standard derivatives: d/dx (xⁿ) = nxⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = –sin x, d/dx (ln x) = 1/x. Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v – uv’)/v².
标准导数：d/dx (xⁿ) = nxⁿ⁻¹，d/dx (sin x) = cos x，d/dx (cos x) = –sin x，d/dx (ln x) = 1/x。乘积法则：(uv)’ = u’v + uv’。商法则：(u/v)’ = (u’v – uv’)/v²。
Chain rule: If y = f(g(x)), then dy/dx = f'(g(x))·g'(x). For implicit differentiation, differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。隐函数微分：方程两边对 x 求导，将 y 视为 x 的函数，然后解出 dy/dx。
Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt). Second derivative: d²y/dx² = d/dt(dy/dx) / (dx/dt).
参数方程求导：若 x = f(t), y = g(t)，则 dy/dx = (dy/dt)/(dx/dt)。二阶导数：d²y/dx² = d/dt(dy/dx) / (dx/dt)。
7. Integration Techniques | 积分技巧
Basic integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ cos x dx = sin x + C, ∫ sin x dx = –cos x + C.
基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1)，∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C，∫ cos x dx = sin x + C，∫ sin x dx = –cos x + C。
Integration by substitution: Let u = g(x), then ∫ f(g(x))g'(x) dx = ∫ f(u) du. For definite integrals, remember to change the limits of integration.
换元积分法：令 u = g(x)，则 ∫ f(g(x))g'(x) dx = ∫ f(u) du。对于定积分，记得更换积分上下限。
Integration by parts: ∫ u dv = uv – ∫ v du. Choose u and dv wisely; the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) often helps prioritise u.
分部积分：∫ u dv = uv – ∫ v du。合理选择 u 和 dv；LIATE 顺序（对数、反三角、代数、三角、指数）通常有助于确定 u 的优先级。
8. Differential Equations | 微分方程
First-order separable ODEs: dy/dx = f(x)g(y). Separate variables: 1/g(y) dy = f(x) dx, then integrate both sides. Don’t forget the constant of integration.
一阶可分离常微分方程：dy/dx = f(x)g(y)。分离变量：1/g(y) dy = f(x) dx，然后两边积分。不要忘记积分常数。
First-order linear ODEs: dy/dx + P(x)y = Q(x). The integrating factor μ = e^(∫ P dx). Multiply through by μ, then the left side becomes d/dx(μ y). Integrate both sides and solve for y.
一阶线性常微分方程：dy/dx + P(x)y = Q(x)。积分因子 μ = e^(∫ P dx)。方程两边乘 μ，左边化为 d/dx(μ y)，两边积分并解出 y。
Second-order linear homogeneous ODEs with constant coefficients: ay” + by’ + cy = 0. Solve characteristic equation ar² + br + c = 0. For distinct real roots r₁, r₂: y = C₁e^(r₁x) + C₂e^(r₂x); repeated root r: y = (C₁ + C₂x)e^(rx); complex roots α ± iβ: y = e^(αx)(C₁ cos βx + C₂ sin βx).
常系数二阶线性齐次常微分方程：ay” + by’ + cy = 0。解特征方程 ar² + br + c = 0。相异实根 r₁, r₂：y = C₁e^(r₁x) + C₂e^(r₂x)；重根 r：y = (C₁ + C₂x)e^(rx)；复根 α ± iβ：y = e^(αx)(C₁ cos βx + C₂ sin βx)。
9. Parametric Equations | 参数方程
Parametric equations define a curve as x = f(t), y = g(t). To convert to Cartesian form, eliminate the parameter t. Often trigonometric identities (like sin²t + cos²t = 1) are useful.
参数方程用 x = f(t)，y = g(t) 定义曲线。要转化为笛卡尔形式，需消去参数 t。三角恒
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