📚 AQA A-Level Maths Question 9: Trigonometric Integrals and Substitution | AQA A-Level 数学第9题：三角积分与代换法
In AQA A-Level Mathematics, Question 9 often tests your ability to combine trigonometric identities with integration techniques. This practice question focuses on a typical multi-part problem where you must prove an identity, integrate a rational trigonometric expression, and then apply substitution to integrate powers of sine. Mastering this style of question is essential for the Pure Mathematics paper.
在 AQA A-Level 数学中，第9题通常考查你将三角恒等式与积分技巧结合运用的能力。本练习题聚焦一道典型的多步问题：你需要证明一个恒等式、积分一个有理三角表达式，并运用代换法求解正弦高次幂的积分。掌握这类题型对纯数学试卷至关重要。
1. Question Overview | 题目概览
The question below is designed to reflect the level and structure of an AQA Paper 1 question. It covers AQA specification topics such as trigonometric identities, indefinite and definite integration, and integration by substitution.
下面这道题旨在反映 AQA 试卷1 的难度与结构。它涵盖 AQA 考试大纲中的三角恒等式、不定积分与定积分、以及代换积分等主题。
Practice Question 9:
练习题 9：
- (a) Show that sin 2x / (1 + cos 2x) = tan x.
- (a) 证明 sin 2x / (1 + cos 2x) = tan x。
- (b) Hence find ∫ sin 2x / (1 + cos 2x) dx.
- (b) 由此求 ∫ sin 2x / (1 + cos 2x) dx。
- (c) Evaluate ∫ from 0 to π/4 of sin 2x / (1 + cos 2x) dx.
- (c) 计算 ∫ 从 0 到 π/4 的 sin 2x / (1 + cos 2x) dx。
- (d) Use the substitution u = cos x to find ∫ sin³ x dx.
- (d) 用代换 u = cos x 求 ∫ sin³ x dx。
This type of multi-part question is usually worth between 8 and 10 marks in total. It tests routine identity manipulation, recall of standard integrals, evaluation of definite integrals, and careful use of a given substitution.
这类多步题通常总分在 8 到 10 分之间。它考查恒等式的常规变形、标准积分的记忆、定积分的计算以及给定代换的规范使用。
2. Specification Links and Assessment Objectives | 大纲对应与评估目标
In the AQA Pure Mathematics specification, this question links to the integration sections where candidates must integrate using trigonometric identities and substitution. The key assessment objectives are AO1, which focuses on accurate recall and routine procedures, and AO2, which focuses on selecting and applying methods in less familiar contexts.
在 AQA 纯数学大纲中，本题对应“积分”部分，要求考生能够使用三角恒等式和代换法进行积分。核心评估目标是 AO1（侧重准确记忆与常规步骤）和 AO2（侧重在较不熟悉的情境中选择并应用方法）。
For full marks, you need to show clear working, use correct notation, and include constants of integration where needed. The mark scheme rewards structured solutions, so avoid missing the connecting steps between parts.
要拿满分，你需要展示清晰的解题过程、使用正确的符号，并在需要时写上积分常数。评分方案奖励结构化解答，因此不要省略各部分之间的衔接步骤。
3. Trigonometric Identity Refresher | 三角恒等式回顾
Before attempting the proof, recall the double angle identities. These formulas allow you to rewrite products and squares of sine and cosine in forms that are much easier to integrate.
在动手证明之前，请回顾二倍角恒等式。这些公式能将正弦和余弦的乘积与平方改写为更容易积分的形式。
- sin 2x = 2 sin x cos x
- sin 2x = 2 sin x cos x
- cos 2x = cos² x - sin² x
- cos 2x = cos² x - sin² x
- cos 2x = 2 cos² x - 1
- cos 2x = 2 cos² x - 1
- cos 2x = 1 - 2 sin² x
- cos 2x = 1 - 2 sin² x
os 2x = 2 sin² x. These two rearrangements appear frequently in AQA integration questions.
由第三种形式 cos 2x = 2 cos² x - 1，可得 1 + cos 2x = 2 cos² x。由第四种形式 cos 2x = 1 - 2 sin² x，可得 1 - cos 2x = 2 sin² x。这两个变形在 AQA 积分题中经常出现。
Choosing the right form of cos 2x is a key exam skill. For part (a), the form involving cos² x makes the denominator simplify in one line.
选择合适的 cos 2x 形式是一项关键考试技能。对于第(a)部分，涉及 cos² x 的形式能在一行内化简分母。
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