Integration Techniques — Mark Scheme Patterns

Mark Allocation Overview

Question Type	Total Marks	M marks	A marks	B marks
Reduction formula derivation	5–6	3	2–3	0
Reduction formula evaluation	2–3	1	1–2	0
Reduction formula (combined)	7–11	4	3–7	0
Integration by parts	3–4	1	1–2	1
Integration by substitution	3–4	1	1–2	1
Rational function integration	5	2	3	0

Pattern: Reduction Formula Derivation (5–6 marks)

• M1: Split integrand (e.g. $(1-x^2)^{n/2} = (1-x^2)(1-x^2)^{(n-2)/2}$)
• M1: Express $I_n$ as sum of $I_{n-2}$ and another integral
• M1: Integration by parts with correct $u$, $dv$ selection
• A1: Correct working for the additional integral
• A1: Correct recurrence relation

典型扣分

• $u$、$dv$ 选反 → 无法得到递推关系 → 扣 M1
• 忘记处理积分上下限 → 扣 A1

Pattern: Reduction Formula Evaluation (2–3 marks)

• M1: Find boundary value $I_0$ or $I_1$
• A1: Correct iterative application (at least 2 steps shown)
• A1: Correct final value of $I_n$

常见边界值

$I_0 = \int_a^b 1\,dx = b-a$

$I_1 = \int_a^b f(x)\,dx \text{ (直接计算)}$

$\int_0^{\pi/2} \sin^0 x\,dx = \frac{\pi}{2}$

$\int_0^{\pi/2} \sin^1 x\,dx = 1$

Pattern: Integration by Parts (3–4 marks)

• M1: Correct choice of $u$ and $dv$
• A1: Correct $uv - \int v\,du$ setup
• A1: Correct final answer
• B1: Correct integration of $dv$ (sometimes given separately)

Pattern: Integration of Rational Functions (5 marks)

• M1: Set up partial fraction form
• A1: Correct coefficients
• M1: Split into separate integrals
• A1: Correct $\ln$ terms
• A1: Correct $\tan^{-1}$ or other terms

评分细节

步骤	标记	说明
设 $\frac{A}{x-1}+\frac{Bx+C}{x^2+1}$	M1	形式正确
解出 $A=2,B=1,C=0$	A1	系数正确
分成三项积分	M1	拆分方法正确
$3\ln\|x-1\|$	A1	$\ln$ 项正确
$2\tan^{-1}x$	A1	$\tan^{-1}$ 项正确

Follow-Through Rules

• 递推公式推导中，若 $I_n$ 表达式写对但后续计算有算术错误，可 ft
• 边界值 $I_0$ 或 $I_1$ 计算错，后续递推可 ft
• 分部积分中 $u$ 和 $dv$ 选反但计算正确，最多扣 M1