📚 Integration of sin²x and cos²x | 三角平方函数 sin²x 与 cos²x 的积分
In A Level Mathematics, integrating functions like sin²x and cos²x often catches students off guard because they cannot be integrated directly using standard antiderivatives. Instead, these squared trigonometric functions must be rewritten using double-angle identities before integration can be performed smoothly. This article unpacks the core techniques, typical exam applications, and common pitfalls, giving you a clear pathway to full marks on such questions in the Edexcel A Level Maths examination.
在A Level数学考试中，sin²x 和 cos²x这类函数的积分往往会让学生措手不及，因为它们无法直接用标准原函数进行积分。必须先利用倍角恒等式将它们改写，才能顺利求解积分。本文将详细解析核心技巧、常见考题应用以及高频错误，帮助你在Edexcel A Level数学考试中稳稳拿下满分。
1. Why direct integration fails | 为什么无法直接积分
mpted to think the answer is (1/3) sin³x + C, as if it followed the power rule. However, differentiation of (1/3) sin³x gives sin²x cosx, not sin²x, because the chain rule introduces the derivative of the inner sine. Squared trigonometric functions are composite in a different way, and none of the elementary antiderivative formulas cover them directly. Recognising this limitation is the first step to mastering the topic.
看到 ∫ sin²x dx 时，你可能会下意识地认为答案是 (1/3) sin³x + C，好像能直接套用幂法则。但是对 (1/3) sin³x 求导得到的是 sin²x cosx，而不是 sin²x，这是因为链式法则引入了内层正弦的导数。平方三角函数的复合方式不同，基本原函数公式都无法直接覆盖它们。认清这个限制是掌握该主题的第一步。
2. Key trigonometric identities from the formula book | 公式表里的核心三角恒等式
The Edexcel data booklet provides the identities cos 2x ≡ 1 – 2 sin²x and cos 2x ≡ 2 cos²x – 1. By rearranging these, we obtain the fundamental rewriting tools: sin²x ≡ ½ (1 – cos 2x) and cos²x ≡ ½ (1 + cos 2x). These forms convert squared trig functions into simple linear combinations that integrate effortlessly. You do not need to memorise the derivations, but you must be able to apply them fluently under time pressure.
Edexcel 的公式册给出了恒等式 cos 2x ≡ 1 – 2 sin²x 和 cos 2x ≡ 2 cos²x – 1。通过移项，我们可以得到关键的改写工具：sin²x ≡ ½ (1 – cos 2x) 以及 cos²x ≡ ½ (1 + cos 2x)。这些形式将平方三角函数转化为简单的线性组合，积分时毫无阻力。你不需要记住推导过程，但必须能够在时间压力下熟练运用。
| Identity | Rearranged form |
| cos 2x = 1 – 2 sin²x | sin²x = ½ (1 – cos 2x) |
| cos 2x = 2 cos²x – 1 | cos²x = ½ (1 + cos 2x) |
| sec²x = 1 + tan²x | tan²x = sec²x – 1 |
3. Integrating sin²x step by step | 逐步积分 sin²x
To integrate sin²x, replace it with ½(1 – cos 2x). Then split the integral as ∫ ½ dx – Thus, ∫ cos 2x dx = (1/2) sin 2x. Multiplying by ½ gives ¼ sin 2x. The final answer is ½ x – ¼ sin 2x + C. Always remember the constant of integration in indefinite integrals.
积分 sin²x 时，先用 ½(1 – cos 2x) 替换，然后将积分拆分为 ∫ ½ dx – ∫ ½ cos 2x dx。第一部分得到 ½ x，第二部分需要用到标准结果 ∫ cos(Ax) dx = (1/A) sin(Ax) + C。于是 ∫ cos 2x dx = (1/2) sin 2x，再乘以 ½ 得到 ¼ sin 2x。最终答案为 ½ x – ¼ sin 2x + C。不定积分中务必带上积分常数。
∫ sin²x dx = ½ x – ¼ sin 2x + C
4. Integrating cos²x with the same strategy | 用相同的策略积分 cos²x
½ cos 2x dx. This yields ½ x + ¼ sin 2x + C. Notice how the only difference from the sin²x result is the sign between the two terms. Students who confuse the signs often lose easy marks; commit this pair of results to memory or quickly re-derive them using the double-angle identities during the exam.
类似地，cos²x 转换为 ½(1 + cos 2x)。它的积分变为 ∫ ½ dx + ∫ ½ cos 2x dx，得到 ½ x + ¼ sin 2x + C。注意它与 sin²x 结果的唯一区别就是两项之间的符号。混淆符号的学生常常丢掉本可轻易拿到的分数；最好把这对结果牢记在心，或者在考场上利用倍角恒等式快速推导一遍。
∫ cos²x dx = ½ x + ¼ sin 2x + C
5. Handling tan²x and related integrals | 处理 tan²x 及相关积分
n the same set of questions. The formula booklet gives sec²x = 1 + tan²x, so tan²x = sec²x – 1. Since the derivative of tan x is sec²x, the integral ∫ tan²x dx = ∫ (sec²x – 1) dx = tan x – roblems, especially when combined with trigonometric substitution or integration by parts.
虽然 tan²x 不属于正弦或余弦的平方，但它经常出现在同一类题目中。公式册中给出了 sec²x = 1 + tan²x，因此 tan²x = sec²x – 1。因为 tan x 的导数是 sec²x，所以 ∫ tan²x dx = ∫ (sec²x – 1) dx = tan x – x + C。这个小技巧在 Edexcel 的积分题中极为常见，尤其是在结合三角代换或分部积分时。
6. Definite integration with limits | 定积分与上下限处理
When evaluating definite integrals like ∫₀π/4 sin²x dx, first find the antiderivative: ½ x – ¼ sin 2x. Substitute the upper limit: (½ × π/4) – ¼ sin(2 × π/4) = π/8 – ¼ sin(π/2) = π/8 – ¼. At the lower limit 0, both terms are 0. Hence the exact value is π/8 – values and radians; Edexcel often expects answers in terms of π and simplified fractions.
计算定积分如 ∫₀π/4 sin²x dx 时，先写出原函数：½ x – ¼ sin 2x。代入上限：(½ × π/4) – ¼ sin(2 × π/4) = π/8 – ¼ sin(π/2) = π/8 – ¼。下限 0 代入后两项均为 0。因此精确值为 π/8 – ¼。请习惯使用精确值和弧度制；Edexcel 常要求答案以 π 和化简分数的形式呈现。
7. Volume of revolution using sin²x and cos²x | sin²x 和 cos²x 在旋转体体积中的应用
π ∫ y² dx often requires integrating y² = sin²x or y² = cos²x. For example, the curve y = √sin x Always be careful to square the function correctly before applying the double-angle identity, and remember to multiply the final result by π.
在绕 x 轴旋转体的体积计算中，公式 V = π ∫ y² dx 常常要求对 y² = sin²x 或 cos²x 进行积分。例如，曲线 y = √sin x 在区间 [0, π] 上平方后得到 y² = sin x，但如果 y = sin x，平方后 y² = sin²x，此时 ∫ sin²x dx 会直接出现。务必在使用倍角公式之前正确平方函数，并记得将最终结果乘以 π。
8. Common exam pitfall: forgetting the chain rule in reverse | 常见陷阱：反向链式法则运用错误
ecause the derivative of sin³x involves a cos factor, the integral of sin²x must somehow involve a division by cos. These errors arise from misapplying the power rule. Always check by differentiating your result. If differentiating ½ x – returns sin²x, you can be confident it is correct. Self-checking is a powerful exam tool.
一个常见错误是直接写成 ∫ sin²x dx = (1/3)sin³x + C，或者认为因为 sin³x 的导数含有 cos 因子，所以 sin²x 的积分就与除以 cos 有关。这类错误源于误用幂法则。一定要通过求导来验证结果。如果对 ½ x – ¼ sin 2x 求导能得到 sin²x，就可以确信答案正确。自我检验是考场上极为有效的工具。
9. Connecting to integration of sin³x and higher powers | 联系 sin³x、cos³x 及更高次幂的积分
in²x and cos²x, you are ready to tackle odd powers like sin³x by splitting sin³x = sinx (1 – uilding blocks for more complex trigonometric integration, making them essential for the Edexcel Pure Mathematics exam.
掌握 sin²x 和 cos²x 之后，你便可以处理奇次幂如 sin³x，将其拆分为 sin³x = sinx (1 – cos²x) 并使用代换 u = cos x。更高的幂次通常也能借助本文所学的恒等式进行降阶。sin²x 和 cos²x 的积分公式是解决更复杂三角积分的基础，对 Edexcel 纯数考试至关重要。
10. Worked example with mixed functions | 混合函数的完整例题
Integrate f(x) = 3 sin²x + 2 cos²x with respect to x. First, rewrite using identities: 3(½ – ½ cos 2x) + 2(½ + ½ cos 2x) = (3/2 – 3/2 cos 2x) + (1 + cos 2x) = 5/2 – ½ cos 2x. Now integrate term by term: ∫ 5/2 dx = 5x/2, and ∫ –½ cos 2x dx = –½ × (1/2) sin 2x = –¼ sin 2x. Thus, the antiderivative is 5x/2 – his composite style is typical of Edexcel questions that test fluency with the identities.
对 f(x) = 3 sin²x + 2 cos²x 求不定积分。先用恒等式改写：3(½ – ½ cos 2x) + 2(½ + ½ cos 2x) = (3/2 – 3/2 cos 2x) + (1 + cos 2x) = 5/2 – ½ cos 2x。然后逐项积分：∫ 5/2 dx = 5x/2，∫ –½ cos 2x dx = –½ × (1/2) sin 2x = –¼ sin 2x。原函数为 5x/2 – ¼ sin 2x + C。这种混合题型是 Edexcel 考查公式熟悉度的典型方式。
11. Quick summary table for revision | 速查复习表
Keeping a compact reference table helps consolidate memory before the exam. The table below includes the three integrals most likely to appear and their results. Use it for last-minute revision, but ensure you understand the derivation steps rather than relying purely on memorisation.
准备一张简洁的速查表有助于在考前巩固记忆。下表列出最常出现的三个积分及其结果。可以在最后冲刺复习时使用，但请务必理解推导过程，不要只依赖死记硬背。
| Integral | Result |
|---|---|
| ∫ sin²x dx | ½ x – ¼ sin 2x + C |
| ∫ cos²x dx | ½ x + ¼ sin 2x + C |
| ∫ tan²x dx | tan x – x + C |
12. Exam technique and mark allocation | 考试技巧与分值分配
In the Edexcel Pure Mathematics papers, integrating sin²x or cos²x is typically worth 3 to 5 marks. Marks are usually allocated for: (1) correctly using the double-angle identity, (2) integrating the constant and cosine terms accurately, and (3) simplifying and including the constant of integration. Show every step clearly; even if you make a slip in simplification, you can still earn method marks. Where volume of revolution is involved, a final multiplication by π is required and often explicitly rewarded.
在 Edexcel 纯数试卷中，积分 sin²x 或 cos²x 的题目通常占 3 至 5 分。分数通常分配在：(1) 正确使用倍角恒等式，(2) 精确积分常数项和余弦项，(3) 化简并加上积分常数。每一步都要清晰展示；即使化简时出现小失误，仍可能获得方法分。涉及旋转体体积时，最后还要乘以 π，这个步骤往往单独有分。
Published by TutorHao | Pure Mathematics Revision Series | aleveler.com
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