Applying Trigonometric Identities in Integration | 三角恒等式在积分中的应用

📚 Applying Trigonometric Identities in Integration | 三角恒等式在积分中的应用

When integrating expressions involving trigonometric functions, direct integration is often impossible because the integrand is not in a standard form. The key is to rewrite the integrand using trigonometric identities, transforming it into a sum of simpler functions that can be integrated term by term.

在对包含三角函数的表达式进行积分时，往往无法直接积分，因为被积函数不是标准形式。关键在于利用三角恒等式改写被积函数，将其转化为若干更简单函数的和，从而可以逐项积分。

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1. The Core Strategy | 核心策略

The fundamental idea is to replace a product or a power of trigonometric functions with an equivalent expression involving sums or constants. This reduces the complexity of the integrand and makes standard integration rules applicable.

基本思想是用一个包含和式或常数的等价表达式替换三角函数的乘积或幂。这降低了被积函数的复杂度，使标准积分法则得以应用。

Common identities used in integration include:

积分中常用的恒等式包括：

• Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.
• uble-angle identities: sin2θ = 2sinθcosθ, cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
• Product-to-sum identities: 2sinAcosB = sin(A+B) + sin(A−B), etc.

• 毕达哥拉斯恒等式：sin²θ + cos²θ = 1，1 + tan²θ = sec²θ，1 + cot²θ = csc²θ。
• 二倍角恒等式：sin2θ = 2sinθcosθ，cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。
• 积化和差恒等式：2sinAcosB = sin(A+B) + sin(A−B)，等等。

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2. Integrating sin²x and cos²x | 对 sin²x 和 cos²x 积分

The squares of sine and cosine cannot be integrated directly. The half-angle forms derived from the double-angle identity are essential:

正弦和余弦的平方不能直接积分。由二倍角恒等式导出的半角形式至关重要：

sin²x = (1 − cos2x)/2, cos²x = (1 + cos2x)/2

Example: ∫ sin²x dx = ∫ (1 − cos2x)/2 dx = x/2 − sin2x/4 + C.

例如：∫ sin²x dx = ∫ (1 − cos2x)/2 dx = x/2 − sin2x/4 + C。

This technique extends to higher even powers by repeated application, and to odd powers by factoring out one factor.

该技术通过重复应用可推广到更高的偶次幂，而对奇次幂则通过提取一个因子来处理。

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3. Integrating sin³x and cos³x | 对 sin³x 和 cos³x 积分

For odd powers, separate one factor and use the Pythagorean identity.

对于奇次幂，分离一个因子并使用毕达哥拉斯恒等式。

du = cosx dx, then the integral becomes ∫ (1 − u²) du = u − u³/3 + C = sinx − sin³x/3 + C.

例如：∫ cos³x dx = ∫ cos²x · cosx dx = ∫ (1 − sin²x) cosx dx。令 u = sinx，du = cosx dx，则积分变为 ∫ (1 − u²) du = u − u³/3 + C = sinx − sin³x/3 + C。

Similarly, ∫ sin³x dx = ∫ (1 − cos²x) sinx dx = −cosx + cos³x/3 + C.

类似地，∫ sin³x dx = ∫ (1 − cos²x) sinx dx = −cosx + cos³x/3 + C。

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4. Integrating Higher Powers | 更高次幂的积分

For powers like sin⁴x or cos⁴x, apply the half-angle formulas twice.

对于 sin⁴x 或 cos⁴x 这样的幂，需要两次应用半角公式。

mple: ∫ sin⁴x dx = ∫ (sin²x)² dx = ∫ ((1 − cos2x)/2)² dx = 1/4 ∫ (1 − 2cos2x + cos²2x) dx.

例如：∫ sin⁴x dx = ∫ (sin²x)² dx = ∫ ((1 − cos2x)/2)² dx = 1/4 ∫ (1 − 2cos2x + cos²2x) dx。

Then use cos²2x = (1 + cos4x)/2 to finish.

然后使用 cos²2x = (1 + cos4x)/2 完成积分。

The final result is ∫ sin⁴x dx = 3x/8 − sin2x/4 + sin4x/32 + C.

最终结果为 ∫ sin⁴x dx = 3x/8 − sin2x/4 + sin4x/32 + C。

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5. Product-to-Sum Identities | 积化和差恒等式

Products of sine and cosine terms are handled by converting them into sums.

正弦和余弦项的乘积通过转化为和来处理。

2sinAcosB = sin(A+B) + sin(A−B)
2cosAcosB = cos(A+B) + cos(A−B)
2sinAsinB = cos(A−B) − cos(A+B)

Example: ∫ sin3x cos2x dx = 1/2 ∫ [sin5x + sinx] dx = −cos5x/10 − cosx/2 + C.

例如：∫ sin3x cos2x dx = 1/2 ∫ [sin5x + sinx] dx = −cos5x/10 − cosx/2 + C。

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6. Integrating tan²x and cot²x | 对 tan²x 和 cot²x 积分

The identity 1 + tan²x = sec²x provides a direct simplification.

恒等式 1 + tan²x = sec²x 提供了直接简化。

Example: ∫ tan²x dx = ∫ (sec²x − 1) dx = tanx − x + C.

例如：∫ tan²x dx = ∫ (sec²x − 1) dx = tanx − x + C。

Similarly, ∫ cot²x dx = ∫ (csc²x − 1) dx = −cotx − x + C.

类似地，∫ cot²x dx = ∫ (csc²x − 1) dx = −cotx − x + C。

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7. Half-Angle Substitution for Rational Expressions | 有理表达式的半角代换

involving rational functions of sinx and cosx, the substitution t = tan(x/2) is powerful.

对于包含 sinx 和 cosx 的有理函数积分，代换 t = tan(x/2) 非常有效。

sinx = 2t/(1 + t²), cosx = (1 − t²)/(1 + t²), dx = 2dt/(1 + t²)

le: ∫ dx/(1 + cosx) = ∫ [2/(1+t²)] / [1 + (1−t²)/(1+t²)] dt = ∫ dt = t + C = tan(x/2) + C.

例如：∫ dx/(1 + cosx) = ∫ [2/(1+t²)] / [1 + (1−t²)/(1+t²)] dt = ∫ dt = t + C = tan(x/2) + C。

This substitution is especially useful when other identities fail.

当其他恒等式失效时，这种代换尤其有用。

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8. Using identities to simplify before integration | 积分前用恒等式简化

Sometimes the integrand can be simplified using a Pythagorean identity before applying another technique.

有时可以在应用其他技巧之前先用毕达哥拉斯恒等式简化被积函数。

cos²x dx = ∫ (sinx cosx)² dx = ∫ (sin2x/2)² dx = 1/4 ∫ sin²2x dx = 1/4 ∫ (1 − cos4x)/2 dx.

例如： cos²x dx = ∫ (sinx cosx)² dx = ∫ (sin2x/2)² dx = 1/4 ∫ sin²2x dx = 1/4 ∫ (1 − cos4x)/2 dx。

The final answer is x/8 − sin4x/32 + C.

最终答案为 x/8 − sin4x/32 + C。

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9. Definite integrals and symmetry | 定积分与对称性

When evaluating definite integrals, identities can also reveal symmetry. For example, ∫₀^π/2 sin²x dx = ∫₀^π/2 cos²x dx because of the identity sinx = cos(π/2 − x).

在计算定积分时，恒等式也能揭示对称性。例如，∫₀^π/2 sin²x dx = ∫₀^π/2 cos²x dx，因为 sinx = cos(π/2 − x)。

Using the average: sin²x + cos²x = 1, so each integral equals 1/2 ∫₀^π/2 1 dx = π/4.

利用平均值：sin²x + cos²x = 1，因此每个积分都等于 1/2 ∫₀^π/2 1 dx = π/4。

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10. Common exam pitfalls | 常见考试误区

ltiple when applying double-angle formulas. For instance, ∫ cos2x dx = sin2x/2, not sin2x.

学生在应用二倍角公式时经常忘记常数倍数。例如，∫ cos2x dx = sin2x/2，而不是 sin2x。

• Always check whether the power is even or odd before choosing a strategy.
• Remember to add the constant of integration for indefinite integrals.
• When using t = tan(x/2), don't forget the factor 2 in dx.

• 在选择策略前，务必判断幂是偶数还是奇数。
• 对于不定积分，记得加上积分常数。
• 使用 t = tan(x/2) 时，不要忘记 dx 中的因子 2。

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11. Worked exam-style example | 考试风格例题

Evaluate ∫₀^π/4 sin²x cos²x dx.

计算 ∫₀^π/4 sin²x cos²x dx。

Using the double-angle identity, sin²x cos²x = (sin²2x)/4 = (1 − cos4x)/8.

利用二倍角恒等式，sin²x cos²x = (sin²2x)/4 = (1 − cos4x)/8。

Therefore ∫₀^π/4 (1 − cos4x)/8 dx = [x/8 − sin4x/32]₀^π/4 = (π/32 − sinπ/32) − (0 − 0) = π/32.

因此 ∫₀^π/4 (1 − cos4x)/8 dx = [x/8 − sin4x/32]₀^π/4 = (π/32 − sinπ/32) − (0 − 0) = π/32。

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12. Summary | 总结

Trigonometric identities are indispensable tools in integration. Mastery of the Pythagorean, double-angle, and product-to-sum identities allows you to transform almost any product or power of trigonometric functions into an integrable form.

三角恒等式是积分中不可或缺的工具。掌握毕达哥拉斯、二倍角和积化和差恒等式，可以让你将几乎任何三角函数的乘积或幂转化为可积分的形式。

Practice rewriting integrands in multiple ways, and always verify your answer by differentiation.

练习用多种方式改写被积函数，并始终通过求导来验证你的答案。

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