绪论：Pre-U CIE 进阶数学的知识体系
Cambridge Pre-U Further Mathematics（进阶数学）是剑桥国际考试委员会（CIE）为高中阶段学生设计的最高难度数学课程之一。它超越了 A-Level Further Mathematics 的深度，涵盖了复数理论、线性代数、双曲函数、极坐标、微分方程、向量微积分、数值方法以及数学归纳法等核心领域。本文将以上模块的核心公式和定理编成速查手册，帮助考生系统梳理知识体系。
Cambridge Pre-U Further Mathematics is one of the most advanced high-school mathematics courses designed by Cambridge International Examinations (CIE). It goes beyond the depth of A-Level Further Mathematics and covers core domains including complex number theory, linear algebra, hyperbolic functions, polar coordinates, differential equations, vector calculus, numerical methods, and proof by induction. This article compiles the core formulas and theorems from each module into a quick-reference handbook to help candidates systematically review the knowledge framework.
第一章：复数 Complex Numbers
复数的基本形式与运算
复数 z = x + iy，其中 x = Re(z)、y = Im(z)，i² = -1。共轭复数定义为 z* = x – iy。复数的模 |z| = sqrt(x² + y²)，幅角 arg(z) = arctan(y/x)，注意根据象限确定正确值。复数加法和减法按实部和虚部分别进行：(a + bi) + (c + di) = (a + c) + (b + d)i。
A complex number z = x + iy, where x = Re(z), y = Im(z), and i² = -1. The complex conjugate is defined as z* = x – iy. The modulus |z| = sqrt(x² + y²), and the argument arg(z) = arctan(y/x), noting that the quadrant must be considered for the correct value. Addition and subtraction of complex numbers are performed separately on real and imaginary components: (a + bi) + (c + di) = (a + c) + (b + d)i.
复数乘法：(a + bi)(c + di) = (ac – bd) + (ad + bc)i。复数除法使用共轭技巧：z1/z2 = (z1 * z2*)/(z2 * z2*) = (z1 * z2*)/|z2|²。这些运算都可以用极坐标形式大大简化。
Complex multiplication: (a + bi)(c + di) = (ac – bd) + (ad + bc)i. Complex division uses the conjugate trick: z1/z2 = (z1 * z2*)/(z2 * z2*) = (z1 * z2*)/|z2|². These operations are greatly simplified when expressed in polar form.
极坐标形式与棣莫弗定理
复数在极坐标中表示为 z = r(cos θ + i sin θ) = re^(iθ)，其中 r = |z|，θ = arg(z)。欧拉公式 e^(iθ) = cos θ + i sin θ 是连接指数函数和三角函数的桥梁。棣莫弗定理（De Moivre’s Theorem）是 Pre-U 考试的核心工具：(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)，适用于任意整数 n。
A complex number in polar form is expressed as z = r(cos θ + i sin θ) = re^(iθ), where r = |z| and θ = arg(z). Euler’s formula e^(iθ) = cos θ + i sin θ bridges exponential and trigonometric functions. De Moivre’s Theorem is a core tool in Pre-U examinations: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ), valid for any integer n.
棣莫弗定理的应用非常广泛：推导 sin(nθ) 和 cos(nθ) 的多项式展开、求解复数的 n 次根、证明三角恒等式。例如，cos(3θ) = 4cos³θ – 3cosθ，sin(3θ) = 3sinθ – 4sin³θ 都可以通过展开 (cos θ + i sin θ)³ 并分离实部和虚部得到。
The applications of De Moivre’s Theorem are extensive: deriving polynomial expansions of sin(nθ) and cos(nθ), finding n-th roots of complex numbers, and proving trigonometric identities. For example, cos(3θ) = 4cos³θ – 3cosθ and sin(3θ) = 3sinθ – 4sin³θ can both be derived by expanding (cos θ + i sin θ)³ and separating real and imaginary parts.
单位根 Roots of Unity
方程 z^n = 1 的 n 个解称为 n 次单位根，它们均匀分布在复平面上的单位圆上。第 k 个根为 ω_k = e^(2πik/n) = cos(2πk/n) + i sin(2πk/n)，k = 0, 1, 2, …, n-1。若记 ω = e^(2πi/n)，则所有 n 次单位根可以表示为 1, ω, ω², …, ω^(n-1)。
The n solutions to z^n = 1 are called the n-th roots of unity. They are evenly distributed on the unit circle in the complex plane. The k-th root is ω_k = e^(2πik/n) = cos(2πk/n) + i sin(2πk/n), k = 0, 1, 2, …, n-1. If we denote ω = e^(2πi/n), then all n-th roots of unity can be expressed as 1, ω, ω², …, ω^(n-1).
重要性质：所有 n 次单位根的和为零：1 + ω + ω² + … + ω^(n-1) = 0；它们的乘积为 (-1)^(n-1)。若 n 为素数，则每个非平凡根 (ω^k, 1 ≤ k < n) 都是本原 n 次单位根。对于一般的方程 z^n = a，根为 z = a^(1/n) * ω^k，其中 a^(1/n) 是 a 的某个 n 次实根。
Important properties: the sum of all n-th roots of unity is zero: 1 + ω + ω² + … + ω^(n-1) = 0; their product is (-1)^(n-1). If n is prime, then every non-trivial root (ω^k, 1 ≤ k < n) is a primitive n-th root of unity. For the general equation z^n = a, the roots are z = a^(1/n) * ω^k, where a^(1/n) is some real n-th root of a.
第二章：矩阵与线性变换 Matrices and Linear Transformations
矩阵运算基本公式
矩阵加法要求同型矩阵，对应元素相加。矩阵乘法要求 A 的列数等于 B 的行数：(AB)_(ij) = sum(A_(ik) * B_(kj), k=1..n)。注意矩阵乘法不满足交换律：一般 AB ≠ BA。单位矩阵 I 满足 AI = IA = A。矩阵的转置：(A^T)_(ij) = A_(ji)，且 (AB)^T = B^T A^T。
Matrix addition requires matrices of the same dimensions, adding corresponding elements. Matrix multiplication requires that the number of columns of A equals the number of rows of B: (AB)_(ij) = sum(A_(ik) * B_(kj), k=1..n). Note that matrix multiplication is not commutative: generally AB ≠ BA. The identity matrix I satisfies AI = IA = A. Matrix transpose: (A^T)_(ij) = A_(ji), and (AB)^T = B^T A^T.
行列式与逆矩阵
对于 2×2 矩阵 A = [[a, b], [c, d]]，行列式 det(A) = ad – bc。若 det(A) ≠ 0，则 A 可逆，逆矩阵 A^(-1) = (1/det(A)) * [[d, -b], [-c, a]]。对于 3×3 矩阵，使用 Sarrus 法则或按行/列展开计算行列式。行列式的性质：det(AB) = det(A) * det(B)，det(A^T) = det(A)，det(A^(-1)) = 1/det(A)。
For a 2×2 matrix A = [[a, b], [c, d]], the determinant det(A) = ad – bc. If det(A) ≠ 0, then A is invertible with inverse A^(-1) = (1/det(A)) * [[d, -b], [-c, a]]. For 3×3 matrices, use Sarrus’s rule or cofactor expansion along a row/column to compute the determinant. Properties of determinants: det(AB) = det(A) * det(B), det(A^T) = det(A), det(A^(-1)) = 1/det(A).
线性变换的几何意义
Pre-U 考试要求从几何角度理解线性变换。矩阵乘法可视为对向量施加线性变换。关键变换包括：旋转（绕原点逆时针旋转 θ 角的矩阵为 [[cos θ, -sin θ], [sin θ, cos θ]]）、反射（关于直线 y = (tan θ)x 的反射矩阵为 [[cos 2θ, sin 2θ], [sin 2θ, -cos 2θ]]）、缩放和剪切变换。行列式的绝对值代表变换后面积（2D）或体积（3D）的缩放因子。
Pre-U examinations require a geometric understanding of linear transformations. Matrix multiplication can be viewed as applying a linear transformation to vectors. Key transformations include: rotation (anticlockwise rotation by angle θ about the origin: [[cos θ, -sin θ], [sin θ, cos θ]]), reflection (reflection in the line y = (tan θ)x: [[cos 2θ, sin 2θ], [sin 2θ, -cos 2θ]]), scaling, and shear transformations. The absolute value of the determinant represents the area (2D) or volume (3D) scaling factor after transformation.
第三章：双曲函数 Hyperbolic Functions
双曲函数是指数函数 e^x 和 e^(-x) 的组合，是 Pre-U 进阶数学的重要考点。定义如下：sinh x = (e^x – e^(-x))/2，cosh x = (e^x + e^(-x))/2，tanh x = sinh x/cosh x = (e^x – e^(-x))/(e^x + e^(-x))。对应的反双曲函数也需要掌握：arsinh x = ln(x + sqrt(x² + 1))，arcosh x = ln(x + sqrt(x² – 1))，artanh x = (1/2) * ln((1+x)/(1-x))。
Hyperbolic functions are combinations of the exponential functions e^x and e^(-x) and are a major topic in Pre-U Further Mathematics. Definitions: sinh x = (e^x – e^(-x))/2, cosh x = (e^x + e^(-x))/2, tanh x = sinh x/cosh x = (e^x – e^(-x))/(e^x + e^(-x)). The corresponding inverse hyperbolic functions must also be mastered: arsinh x = ln(x + sqrt(x² + 1)), arcosh x = ln(x + sqrt(x² – 1)), artanh x = (1/2) * ln((1+x)/(1-x)).
双曲函数满足一系列类似于三角函数的恒等式（Osborn’s Rule：把三角恒等式中的 sin 换成 sinh、cos 换成 cosh，每当出现 sinh 的乘积项时，改变该项的符号即可衍生出对应的双曲恒等式）。核心恒等式：cosh²x – sinh²x = 1；sinh(2x) = 2sinh x cosh x；cosh(2x) = cosh²x + sinh²x = 2cosh²x – 1 = 1 + 2sinh²x。
Hyperbolic functions satisfy a series of identities analogous to trigonometric ones (Osborn’s Rule: replace sin with sinh and cos with cosh in a trigonometric identity, and whenever a product of two sinh terms appears, change the sign of that term). Core identities: cosh²x – sinh²x = 1; sinh(2x) = 2sinh x cosh x; cosh(2x) = cosh²x + sinh²x = 2cosh²x – 1 = 1 + 2sinh²x.
微积分方面：d/dx(sinh x) = cosh x；d/dx(cosh x) = sinh x；d/dx(tanh x) = sech²x。积分公式：∫sinh x dx = cosh x + C；∫cosh x dx = sinh x + C；∫tanh x dx = ln(cosh x) + C。在使用反双曲函数进行积分时，∫1/sqrt(x² + a²) dx = arsinh(x/a) + C 和 ∫1/sqrt(x² – a²) dx = arcosh(x/a) + C 是重要公式。
In calculus: d/dx(sinh x) = cosh x; d/dx(cosh x) = sinh x; d/dx(tanh x) = sech²x. Integration formulas: ∫sinh x dx = cosh x + C; ∫cosh x dx = sinh x + C; ∫tanh x dx = ln(cosh x) + C. When using inverse hyperbolic functions in integration, ∫1/sqrt(x² + a²) dx = arsinh(x/a) + C and ∫1/sqrt(x² – a²) dx = arcosh(x/a) + C are important formulas.
第四章：极坐标 Polar Coordinates
极坐标是描述平面上点位置的另一种方式，在 Pre-U 考试中经常与积分结合。平面上一点由极径 r（到原点的距离）和极角 θ（与正 x 轴的夹角）确定。直角坐标与极坐标的转换关系：x = r cos θ，y = r sin θ；r = sqrt(x² + y²)，θ = arctan(y/x)。极坐标曲线的方程通常写为 r = f(θ)。
Polar coordinates provide an alternative way to describe the position of a point in a plane and frequently appear in Pre-U exams combined with integration. A point is defined by the radial distance r (distance from the origin) and the polar angle θ (angle measured from the positive x-axis). Conversion between Cartesian and polar coordinates: x = r cos θ, y = r sin θ; r = sqrt(x² + y²), θ = arctan(y/x). Polar curves are typically expressed as r = f(θ).
在极坐标下计算曲线所围面积的核心公式：A = (1/2)∫[α to β] r² dθ。这一公式源自将区域分割为无数个面积约为 (1/2)r²Δθ 的扇形。常见极坐标曲线：心形线 r = a(1 + cos θ)、三叶玫瑰线 r = a sin(3θ)、以及阿基米德螺线 r = aθ。
The core formula for calculating the area enclosed by a polar curve is: A = (1/2)∫[α to β] r² dθ. This formula arises from partitioning the region into infinitely many sectors of approximate area (1/2)r²Δθ. Common polar curves include: the cardioid r = a(1 + cos θ), the three-petalled rose r = a sin(3θ), and the Archimedean spiral r = aθ.
第五章：微分方程 Differential Equations
一阶微分方程
可分离变量方程是 Pre-U 考试中最基础的微分方程类型，形式为 dy/dx = f(x)g(y)。解法：将变量分离为 (1/g(y))dy = f(x)dx，然后两边积分。例如：dy/dx = xy 可重写为 (1/y)dy = x dx，积分得 ln|y| = x²/2 + C，即 y = Ae^(x²/2)。
Separable equations are the most fundamental type of differential equation in Pre-U exams, taking the form dy/dx = f(x)g(y). Solution method: separate variables to obtain (1/g(y))dy = f(x)dx, then integrate both sides. Example: dy/dx = xy can be rewritten as (1/y)dy = x dx, integrating to give ln|y| = x²/2 + C, i.e. y = Ae^(x²/2).
一阶线性微分方程的标准形式为 dy/dx + P(x)y = Q(x)。使用积分因子法求解：积分因子 IF = e^(∫P(x)dx)，将方程两边同时乘以积分因子，左侧即变为 d/dx(y * IF)。因此通解为 y * IF = ∫Q(x) * IF dx + C。例如：dy/dx + 2xy = x 的积分因子为 e^(x²)。
The standard form of a first-order linear differential equation is dy/dx + P(x)y = Q(x). Solve using the integrating factor method: IF = e^(∫P(x)dx), multiply both sides by the integrating factor, and the left side becomes d/dx(y * IF). Therefore the general solution is y * IF = ∫Q(x) * IF dx + C. Example: for dy/dx + 2xy = x, the integrating factor is e^(x²).
二阶线性微分方程
二阶常系数齐次线性微分方程的形式为 a * d²y/dx² + b * dy/dx + c * y = 0。解的特性由特征方程 a * m² + b * m + c = 0 的根决定。若两根为不等实根 m1 和 m2，通解为 y = A * e^(m1*x) + B * e^(m2*x)；若为相等实根 m，通解为 y = (A + Bx) * e^(mx)；若为共轭复根 α ± iβ，通解为 y = e^(αx) * (A cos βx + B sin βx)。
The form of a second-order homogeneous linear differential equation with constant coefficients is a * d²y/dx² + b * dy/dx + c * y = 0. The nature of the solution is determined by the roots of the characteristic equation a * m² + b * m + c = 0. For distinct real roots m1 and m2: y = A * e^(m1*x) + B * e^(m2*x). For repeated real root m: y = (A + Bx) * e^(mx). For complex conjugate roots α ± iβ: y = e^(αx) * (A cos βx + B sin βx).
对于非齐次方程 a * d²y/dx² + b * dy/dx + c * y = f(x)，通解 = 齐次方程的通解 + 特解。求特解的方法取决于 f(x) 的具体形式：若 f(x) 是多项式，设 y_p 为同次多项式；若 f(x) = ke^(px)，设 y_p = λe^(px)（需要考虑是否与齐次解共振）；若 f(x) = k cos(px) 或 k sin(px)，设 y_p = λ cos(px) + μ sin(px)。
For the non-homogeneous equation a * d²y/dx² + b * dy/dx + c * y = f(x), the general solution = complementary function + particular integral. The method for finding the particular integral depends on the form of f(x): if f(x) is a polynomial, try y_p as a polynomial of the same degree; if f(x) = ke^(px), try y_p = λe^(px) (watching for resonance with the complementary function); if f(x) = k cos(px) or k sin(px), try y_p = λ cos(px) + μ sin(px).
第六章：麦克劳林级数与泰勒级数 Maclaurin and Taylor Series
麦克劳林级数将函数在 x = 0 附近展开为幂级数：f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … 。泰勒级数是其推广，在 x = a 处展开：f(x) = f(a) + f'(a)(x-a) + f”(a)(x-a)²/2! + …。
The Maclaurin series expands a function as a power series about x = 0: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … . The Taylor series is its generalization, expanding about x = a: f(x) = f(a) + f'(a)(x-a) + f”(a)(x-a)²/2! + … .
Pre-U 考试要求熟记以下重要级数：e^x = 1 + x + x²/2! + x³/3! + … （收敛于所有实数）；sin x = x – x³/3! + x⁵/5! – … ；cos x = 1 – x²/2! + x⁴/4! – … ；ln(1+x) = x – x²/2 + x³/3 – x⁴/4 + … （收敛于 -1 < x ≤ 1）；(1+x)^n = 1 + nx + n(n-1)x²/2! + ... （二项式级数，收敛于 |x| < 1）。
Pre-U exams require memorisation of the following important series: e^x = 1 + x + x²/2! + x³/3! + … (convergent for all real x); sin x = x – x³/3! + x⁵/5! – … ; cos x = 1 – x²/2! + x⁴/4! – … ; ln(1+x) = x – x²/2 + x³/3 – x⁴/4 + … (convergent for -1 < x ≤ 1); (1+x)^n = 1 + nx + n(n-1)x²/2! + ... (binomial series, convergent for |x| < 1).
第七章：向量微积分 Vector Calculus
Pre-U 进阶数学引入三维空间中的向量运算，包括点积（dot product）和叉积（cross product）。对于向量 a = a1i + a2j + a3k 和 b = b1i + b2j + b3k，点积为 a·b = a1b1 + a2b2 + a3b3 = |a||b| cos θ，其中 θ 为两向量夹角。叉积 a×b 是一个向量，其方向垂直于 a 和 b 所在的平面，模 |a×b| = |a||b| sin θ。
Pre-U Further Mathematics introduces vector operations in three-dimensional space, including the dot product and cross product. For vectors a = a1i + a2j + a3k and b = b1i + b2j + b3k, the dot product is a·b = a1b1 + a2b2 + a3b3 = |a||b| cos θ, where θ is the angle between the vectors. The cross product a×b is a vector perpendicular to the plane containing a and b, with magnitude |a×b| = |a||b| sin θ.
叉积的计算公式（使用行列式）：a×b = |i j k ; a1 a2 a3 ; b1 b2 b3| = (a2b3 – a3b2)i + (a3b1 – a1b3)j + (a1b2 – a2b1)k。叉积是反交换的：a×b = -(b×a)。三重标量积 a·(b×c) 表示以 a、b、c 为棱的平行六面体的体积，其值为对应 3×3 矩阵的行列式。
The cross product formula (using a determinant): a×b = |i j k ; a1 a2 a3 ; b1 b2 b3| = (a2b3 – a3b2)i + (a3b1 – a1b3)j + (a1b2 – a2b1)k. The cross product is anti-commutative: a×b = -(b×a). The scalar triple product a·(b×c) represents the volume of the parallelepiped with edges a, b, c, and its value is the determinant of the corresponding 3×3 matrix.
空间直线的向量方程为 r = a + t d，其中 a 是直线上一点的位置向量，d 是方向向量。空间平面的向量方程为 r·n = d（用法向量和常数表示），或者 r = a + s u + t v（用两个方向向量参数化）。求两平面交线、点到平面的距离（|ax1+by1+cz1-d|/sqrt(a²+b²+c²)）等也是考试重点。
The vector equation of a line in 3D space is r = a + t d, where a is the position vector of a point on the line and d is the direction vector. The vector equation of a plane is r·n = d (using the normal vector and a constant), or r = a + s u + t v (parameterised with two direction vectors). Finding the line of intersection of two planes, the distance from a point to a plane (|ax1+by1+cz1-d|/sqrt(a²+b²+c²)), and related problems are also key exam topics.
第八章：数值方法 Numerical Methods
在无法获得解析解的情况下，Pre-U 考试要求考生熟悉并应用数值方法。牛顿-拉弗森法（Newton-Raphson Method）用于逼近方程 f(x) = 0 的根：x_(n+1) = x_n – f(x_n)/f'(x_n)。该方法收敛速度极快（二次收敛），但需要好的初始猜测值，且在 f'(x) 接近零的点可能发散。
When analytical solutions are unavailable, Pre-U exams require candidates to be familiar with and apply numerical methods. The Newton-Raphson method approximates roots of f(x) = 0: x_(n+1) = x_n – f(x_n)/f'(x_n). This method converges extremely rapidly (quadratic convergence), but requires a good initial guess and may diverge near points where f'(x) is close to zero.
数值积分方面，梯形法则（Trapezium Rule）将积分区间等分为 n 个条形，每个条形用梯形面积近似：∫[a to b] f(x)dx ≈ (h/2)[y0 + 2(y1 + y2 + … + y_(n-1)) + yn]，其中 h = (b-a)/n。辛普森法则（Simpson’s Rule）使用抛物线近似，精度更高但要求 n 为偶数：∫[a to b] f(x)dx ≈ (h/3)[y0 + 4(y1 + y3 + …) + 2(y2 + y4 + …) + yn]。
For numerical integration, the Trapezium Rule divides the integration interval into n equal strips, approximating each as a trapezoid: ∫[a to b] f(x)dx ≈ (h/2)[y0 + 2(y1 + y2 + … + y_(n-1)) + yn], where h = (b-a)/n. Simpson’s Rule uses parabolic approximations, offering higher accuracy but requiring n to be even: ∫[a to b] f(x)dx ≈ (h/3)[y0 + 4(y1 + y3 + …) + 2(y2 + y4 + …) + yn].
第九章：数学归纳法 Proof by Induction
数学归纳法是 Pre-U 进阶数学中最核心的证明方法之一，用来证明对所有正整数 n 成立的命题。其标准结构分为四步：(1) 基础步骤 – 验证 n = 1（或最小有效值）时命题成立；(2) 归纳假设 – 假设 n = k 时命题成立；(3) 归纳步骤 – 利用该假设证明 n = k+1 时命题也成立；(4) 结论 – 根据归纳原理，命题对所有正整数 n 成立。
Proof by induction is one of the most central proof techniques in Pre-U Further Mathematics, used to prove propositions that hold for all positive integers n. Its standard structure has four steps: (1) Base case – verify the proposition holds for n = 1 (or the smallest valid value); (2) Induction hypothesis – assume the proposition holds for n = k; (3) Induction step – use this hypothesis to prove the proposition holds for n = k+1; (4) Conclusion – by the principle of induction, the proposition holds for all positive integers n.
考试中最常见的归纳题型包括：求和公式证明（如证明 1² + 2² + … + n² = n(n+1)(2n+1)/6）、整除性证明（如 n³ – n 整除 3）、矩阵幂次证明（如证明特定 2×2 矩阵的 n 次幂的公式）、以及递归数列性质的证明。在整除性证明中，关键的技巧是将 n = k+1 的表达式分解为归纳假设部分的倍数加上另一个显然整除的部分。
The most common induction problems in exams include: proving summation formulas (e.g. 1² + 2² + … + n² = n(n+1)(2n+1)/6), proving divisibility (e.g. n³ – n is divisible by 3), proving matrix powers (e.g. deriving a formula for the n-th power of a specific 2×2 matrix), and proving properties of recursively defined sequences. In divisibility proofs, the key technique is to decompose the expression for n = k+1 into a multiple of the induction hypothesis part plus another obviously divisible part.
第十章：进阶积分技术 Advanced Integration Techniques
Pre-U 进阶数学的积分部分远超出了 A-Level 的范畴。除了基本积分方法（代换法、分部积分法、部分分式法）外，考生还需要掌握三角代换和双曲代换。对于 ∫1/sqrt(a² – x²) dx，令 x = a sin θ；对于 ∫1/sqrt(a² + x²) dx，令 x = a sinh u（或者使用反正弦双曲函数的公式）；对于 ∫1/sqrt(x² – a²) dx，令 x = a cosh u。
The integration section in Pre-U Further Mathematics extends well beyond A-Level. Beyond basic integration methods (substitution, integration by parts, partial fractions), candidates must also master trigonometric and hyperbolic substitutions. For ∫1/sqrt(a² – x²) dx, substitute x = a sin θ; for ∫1/sqrt(a² + x²) dx, substitute x = a sinh u (or use the inverse hyperbolic sine formula); for ∫1/sqrt(x² – a²) dx, substitute x = a cosh u.
分部积分法（Integration by Parts）的公式是 ∫u dv = uv – ∫v du。在涉及 ln x、反三角函数、或需要重复应用分部积分的情况（如 ∫e^x sin x dx，需要分部积分两次然后解出原积分）中，合理选择 u 和 dv 至关重要。LIATE 法则（Logarithmic > Inverse trig > Algebraic > Trigonometric > Exponential）有助于确定优先设为 u 的函数。
The integration by parts formula is ∫u dv = uv – ∫v du. In cases involving ln x, inverse trigonometric functions, or repeated application (e.g. ∫e^x sin x dx, requiring integration by parts twice and then solving for the original integral), careful choice of u and dv is crucial. The LIATE rule (Logarithmic > Inverse trig > Algebraic > Trigonometric > Exponential) helps determine which function to set as u.
部分分式分解（Partial Fractions）用于有理函数的积分。对于分母可分解为不同线性因子的情况：1/[(x-a)(x-b)] = A/(x-a) + B/(x-b)。对于重复线性因子：1/(x-a)² 对应 A/(x-a) + B/(x-a)²。对于不可再分解的二次因子（分母含 ax²+bx+c 且判别式 < 0）：对应项的形式为 (Ax+B)/(ax²+bx+c)。熟练掌握覆盖法（Cover-Up Method）可以在考试中节省大量时间。
Partial fraction decomposition is used for integrating rational functions. For denominators factorisable into distinct linear factors: 1/[(x-a)(x-b)] = A/(x-a) + B/(x-b). For repeated linear factors: 1/(x-a)² corresponds to A/(x-a) + B/(x-a)². For irreducible quadratic factors (denominator containing ax²+bx+c with discriminant < 0): the corresponding term takes the form (Ax+B)/(ax²+bx+c). Mastering the Cover-Up Method saves substantial time in exams.
总结：Pre-U CIE 进阶数学备考策略
Pre-U CIE Further Mathematics 是一门要求极高的课程，其广度跨越了从抽象代数到应用微积分的整个数学光谱。本文梳理了十个核心模块的关键公式和定理，但要真正掌握这门学科，单纯记忆公式是不够的。高效的备考策略应当是：建立模块间的内在联系（例如复数与三角函数、双曲函数与指数函数、矩阵与线性变换的几何直观），通过历年真题反复练习应用能力，并养成规范性书写的习惯以确保在证明题中获得满分。预祝各位考生在 Pre-U 考试中取得理想成绩！
Pre-U CIE Further Mathematics is an exceptionally demanding course whose breadth spans the entire mathematical spectrum from abstract algebra to applied calculus. This article has organised the key formulas and theorems across ten core modules. However, to truly master this subject, mere formula memorisation is insufficient. An effective revision strategy should be: build interconnections between modules (e.g. complex numbers and trigonometry, hyperbolic functions and exponentials, geometric intuition for matrices and linear transformations), repeatedly practise application skills through past papers, and develop rigorous writing habits to secure full marks on proof questions. Best wishes to all candidates for success in the Pre-U examinations!