📚 PDF Joiner (4) – Page 207: Trigonometric Differentiation & Integration | PDF合并(4) – 第207页：三角函数微分与积分
Page 207 of the PDF Joiner (4) compilation brings together a focused set of A-Level style questions that test your ability to differentiate and integrate trigonometric functions. Whether it’s applying the chain rule to sin(2x), tackling products like x cos x, or reversing the process to find an antiderivative, this page is a checkpoint for fluency. In this article, we will unpack the core techniques, provide step-by-step reasoning, and ensure you can handle any trigonometric calculus problem that Edexcel might throw at you.
PDF合并(4)的第207页汇集了一组A-Level风格的题目，专门考察对三角函数进行微分和积分的能力。无论是将链式法则应用于 sin(2x)、处理 x cos x 这样的乘积，还是逆向操作求原函数，本页都是一道熟练度的关卡。在本文中，我们将剖析核心技术，提供逐步推理，并确保你能应对爱德思可能出现的任何三角微积分问题。
1. Core Derivatives of Sine and Cosine | 正弦与余弦的基本导数
The foundation of all trigonometric differentiation lies in two results: d/dx (sin x) = cos x and d/dx (cos x) = −sin x. These are given in the Edexcel formula booklet, but you must know them by heart to avoid sign errors in the exam. The derivative of sin x is cos x, and because cosine decreases as x increases, its derivative is negative sine. Always check the sign when you differentiate cos — it flips to negative.
所有三角函数微分的基础在于两个结果：d/dx (sin x) = cos x 和 d/dx (cos x) = −sin x。爱德思的公式手册中会给出它们，但你仍需熟记，以免在考试中出现符号错误。sin x 的导数是 cos x，而由于 cos x 随着 x 增大而减小，其导数为负的正弦。对 cos 求导时务必检查符号——它会变为负值。
2. Differentiating the Tangent Function | 正切函数的微分
Although tan x can be differentiated as sin x / cos x using the quotient rule, Edexcel candidates are expected to recall that d/dx (tan x) = sec² x. This result can also be written as 1 / cos² x. It appears frequently when the chain rule comes into play with arguments like tan(5x) or tan(θ/2). Sec² x is always positive, so tan x is strictly increasing on each defined interval.
虽然 tan x 可以利用商法则将其视为 sin x / cos x 进行微分，但爱德思考生应牢记 d/dx (tan x) = sec² x。该结果也可写作 1 / cos² x。当链式法则与 tan(5x) 或 tan(θ/2) 等参数结合时，它频繁出现。sec² x 恒为正，因此 tan x 在每个有定义的区间上严格递增。
3. The Chain Rule with Trigonometric Functions | 三角函数的链式法则
When the angle is more than just x, the chain rule scales the derivative by the coefficient of x. For instance, differentiate y = sin(3x): the outside derivative cos(3x) is multiplied by the derivative of the inside, 3, giving 3 cos(3x). Similarly, d/dx [cos(πx)] = −π sin(πx). If the argument is a function like x², the chain rule demands extra care: d/dx [sin(x²)] = 2x cos(x²).
当角度不仅仅是 x 时，链式法则会将导数乘以 x 的系数。例如，对 y = sin(3x) 求导：外部导数 cos(3x) 乘以内层函数 3 的导数，得到 3 cos(3x)。类似地，d/dx [cos(πx)] = −π sin(πx)。若参数是类似 x² 的函数，链式法则要求格外小心：d/dx [sin(x²)] = 2x cos(x²)。
4. Product and Quotient Rules Involving Trig | 涉及三角函数的乘积与商法则
Questions on page 207 often combine trigonometric functions with algebraic terms. For y = x² sin x, use the product rule: u = x² → u’ = 2x, v = sin x → v’ = cos x. Then dy/dx = 2x sin x + x² cos x. If the function is a quotient like (sin x) / x, apply the quotient rule: the derivative is [x cos x − sin x] / x². Write out u, v, u’, v’ systematically to minimise errors.
第207页的题目常将三角函数与代数项组合在一起。对于 y = x² sin x，使用乘积法则：u = x² → u’ = 2x, v = sin x → v’ = cos x。于是 dy/dx = 2x sin x + x² cos x。若函数为 (sin x) / x 这样的商，则应用商法则：导数为 [x cos x − sin x] / x²。系统地写出 u, v, u’, v’ 可最大程度减少错误。
5. Second Derivatives and Trigonometric Expressions | 二阶导数与三角表达式
Edexcel often asks for the second derivative d²y/dx² to determine concavity or verify a differential equation. If y = cos(2x), the first derivative is −2 sin(2x). Differentiating again gives −4 cos(2x), which is −4y. This relationship is characteristic of simple harmonic motion. Always simplify the first derivative before moving to the second to avoid algebraic clutter.
爱德思常要求计算二阶导数 d²y/dx²，以确定凹凸性或验证微分方程。若 y = cos(2x)，一阶导数为 −2 sin(2x)。再次求导得到 −4 cos(2x)，即 −4y。这种关系是简谐运动的特征。务必先简化一阶导数再进行二阶求导，避免代数繁乱。
6. Basic Integration of Sine and Cosine | 正弦与余弦的基本积分
Integration reverses differentiation: ∫ sin x dx = −cos x + C, and ∫ cos x dx = sin x + C. The negative sign with sine often catches students out — think of it as ‘differentiate cos gives −sin, so integrate sin gives −cos’. For definite integrals, remember to apply limits carefully, especially when the result involves negative cosine values.
积分是微分的逆运算：∫ sin x dx = −cos x + C，而 ∫ cos x dx = sin x + C。正弦积分时的负号常让学生出错——可以这样想：’对 cos 求导得 −sin，所以对 sin 积分得 −cos’。对于定积分，务必仔细代入上下限，尤其是结果涉及负的余弦值时。
7. Reverse Chain Rule for Trigonometric Integrals | 三角积分的逆向链式法则
When integrating a composite trig function like cos(4x), divide by the coefficient of x. So ∫ cos(4x) dx = (1/4) sin(4x) + C. For ∫ sin(3x − 1) dx, the antiderivative is −(1/3) cos(3x − 1) + C. This ‘reverse chain rule’ only works when the inner function is linear. If the inner function is non-linear, such as cos(x²), the antiderivative cannot be expressed in elementary forms — a point worth noting for Edexcel.
当对复合三角函数如 cos(4x) 积分时，需除以 x 的系数。因此 ∫ cos(4x) dx = (1/4) sin(4x) + C。对于 ∫ sin(3x − 1) dx，其原函数为 −(1/3) cos(3x − 1) + C。这种’逆向链式法则’仅在内层函数为线性时有效。若内层函数非线性，例如 cos(x²)，原函数无法用初等形式表达——这一点值得爱德思考生注意。
8. Integrating Powers of Trig Functions Using Identities | 利用恒等式对三角函数的幂次积分
To integrate sin² x or cos² x, you must use the double-angle identities: sin² x = ½(1 − cos 2x) and cos² x = ½(1 + cos 2x). This converts the square into a simple cosine term that can be integrated using the reverse chain rule. For ∫ sin² x dx, the result is (1/2)x − (1/4) sin 2x + C. Edexcel examiners expect you to choose the correct identity and handle the factor of ½ accurately.
要对 sin² x 或 cos² x 积分，必须使用倍角恒等式：sin² x = ½(1 − cos 2x) 及 cos² x = ½(1 + cos 2x)。这将平方项转化为可用逆向链式法则积分的简单余弦项。对于 ∫ sin² x dx，结果是 (1/2)x − (1/4) sin 2x + C。爱德思考官期望你能选择正确的恒等式并准确处理因子 ½。
9. Trigonometric Integrals Leading to Logarithms | 导出对数的三角积分
Recognising the form ∫ f'(x)/f(x) dx = ln |f(x)| + C is crucial. When the integrand is tan x, write it as sin x / cos x. The numerator sin x is the negative derivative of the denominator cos x (since derivative of cos x is −sin x). Thus ∫ tan x dx = −ln |cos x| + C, which is often written as ln |sec x| + C. Similarly, ∫ cot x dx = ln |sin x| + C.
识别形式 ∫ f'(x)/f(x) dx = ln |f(x)| + C 至关重要。当被积函数为 tan x 时，可写为 sin x / cos x。分子 sin x 是分母 cos x 导数的相反数（因为 cos x 的导数为 −sin x）。因此 ∫ tan x dx = −ln |cos x| + C，常写作 ln |sec x| + C。类似地，∫ cot x dx = ln |sin x| + C。
10. Mixed Application Problems | 综合应用问题
Page 207 closes with problems that blend differentiation and integration with kinematics or area under a curve. For example, a particle’s velocity is given by v = 2 sin t + t cos t. To find acceleration, differentiate v with respect to t. To find displacement, integrate v. Structured practice with these mixed scenarios reinforces why mastering trigonometric calculus is essential — not only for pure mathematics but also for mechanics sections.
第207页的最后是融合微分、积分与运动学或曲线下面积的问题。例如，一质点的速度由 v = 2 sin t + t cos t 给出。为求加速度，对 v 关于 t 求导。为求位移，对 v 积分。通过这些混合情景的结构化练习，可以强化为何精通三角微积分至关重要——无论是纯数学还是力学部分。
11. Common Pitfalls and How to Avoid Them | 常见陷阱及避免方法
Sign errors are the most frequent mistake: watch the negative when differentiating cos or integrating sin. Forgetting the chain factor when integrating sin(kx) or cos(kx) leads to lost marks. When using identities, always double-check the half-angle constants. Finally, in definite integrals, check whether the function is odd or even over a symmetric interval — for instance, ∫[−a, a] sin x dx = 0 because sin is odd. These small checks save valuable time.
符号错误是最常见的失误：在对 cos 求导或对 sin 积分时注意负号。对 sin(kx) 或 cos(kx) 积分时忘记链式因子会导致丢分。使用恒等式时，务必核对半角常数。最后，在定积分中，检查对称区间上函数的奇偶性——例如，∫[−a, a] sin x dx = 0，因为 sin 是奇函数。这些微小的检查能节省宝贵的时间。
12. Exam Technique: Presenting Your Work Clearly | 应试技巧：清晰呈现解题过程
Edexcel examiners stress method marks. When differentiating, explicitly state the rule you are using. When integrating, show the step where you divide by the coefficient of x. If you use a trigonometric substitution or identity, write it down. Even if the final answer is incorrect, a clear method will secure most marks. Use the formula booklet wisely but do not rely on it for basic derivatives — it will slow you down.
爱德思考官注重方法分。求导时，明确说明所使用的法则。积分时，展示除以 x 系数的步骤。若使用三角代换或恒等式，请写下来。即使最终答案错误，清晰的解题过程也能确保大部分分数。合理利用公式手册，但不要在基本导数上依赖它——这会拖慢你的速度。
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