Integration Techniques - Chapter 14 | 积分技巧 第14章

📚 Integration Techniques - Chapter 14 | 积分技巧 第14章

Due to a potential misinput in the requested topic, this article addresses a standard IB Mathematics: Analysis and Approaches or Applications and Interpretation chapter on Integration Techniques, often covered around Chapter 14. This content adheres to the specified bilingual educational format and focuses on developing core calculus skills such as substitution, integration by parts, and their applications. No inappropriate material will be discussed.

鉴于请求选题可能存在误输入，本文聚焦于IB数学分析与方法或应用与解释中常在第14章涉及的积分技巧。内容遵循指定的双语教育格式，重点培养学生核心微积分技能，如换元积分法、分部积分法及其应用。不讨论任何不适当的内容。

1. Review of Indefinite Integration | 不定积分复习

amily of antiderivatives of a function f(x). The notation is ∫ f(x) dx = F(x) + C, where F' = f(x) and C is an arbitrary constant. Basic power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ –1.

在掌握高级技巧之前，回顾一下不定积分表示函数 f(x) 的原函数族。记作 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)，C 为任意常数。基本幂函数规则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，当 n ≠ –1 时成立。

2. Integration by Substitution – The Chain Rule in Reverse | 换元积分法 – 反向链式法则

part of the chain rule for differentiation. If we have an integral of the form ∫ f(g(x))·g'(x) dx, we set u = g(x), so du = g' ften simpler. Always remember to substitute back to x at the end for indefinite integrals.

换元积分法是微分链式法则的直接逆运算。如果遇到形如 ∫ f(g(x))·g'(x) dx 的积分，我们设 u = g(x)，从而 du = g'(x) dx。积分转化为 ∫ f(u) du，通常更简单。对于不定积分，最后务必用 x 替换回 u。

3. Definite Integrals and Substitution – Adjusting Limits | 定积分与换元 – 调整积分限

en cleaner because you do not need to revert to the original variable. Given ∫ₐᵇ f(g(x))·g'(x) dx, let u = g(x), then du = g'(x) dx. Change the limits: when x = a, u = g(a); when x = b, u = g(b). Then evaluate ∫_g(a)^g(b) f(u) du directly. No need to return to x.

对于定积分，换元法更加简洁，因为你不需要还原变量。给定 ∫ₐᵇ f(g(x))·g'(x) dx，令 u = g(x)，则 du = g'(x) dx。改变积分限：当 x = a 时，u = g(a)；当 x = b 时，u = g(b)。然后直接计算 ∫_g(a)^g(b) f(u) du，无需再回到 x。

4. Recognizing Substitution Patterns – Linear Argument and More | 识别换元模式 – 线性内函及其他

, or dx = du/a. Then the integral becomes (1/a) ∫ f(u) du. For example, ∫ sin(3x + 2) dx = – for integrals where a function and its derivative appear together, such as ∫ (ln x)/x dx.

常见模式是 ∫ f(ax + b) dx。设 u = ax + b，则 du = a dx，即 dx = du/a。积分变为 (1/a) ∫ f(u) du。例如，∫ sin(3x + 2) dx = –(1/3) cos(3x + 2) + C。同时留意函数与其导数同时出现的情况，如 ∫ (ln x)/x dx。

5. Integration by Parts – The Product Rule Reversed | 分部积分法 – 反向乘积法则

Integration by parts derives from the product rule: (uv)' = u'v + uv'. Rearranging, ∫ uv' dx = uv – ∫ u'v dx. The formula is often written as ∫ u dv = uv – igonometric, Exponential); the rest is dv. This technique is crucial for products like x·eˣ or x·sin x.

分部积分法源自乘积求导法则：(uv)' = u'v + uv'。移项得 ∫ uv' dx = uv – ∫ u'v dx。公式通常写作 ∫ u dv = uv – ∫ v du。策略：按照 LIATE 规则选择 u（对数、反三角、代数、三角、指数函数），剩余部分为 dv。此法对于 x·eˣ 或 x·sin x 等乘积形式至关重要。

6. Repeated Integration by Parts and Tabular Method | 多次分部积分与表格法

s integration by parts must be applied several times, especially for integrals like ∫ x² eˣ dx or ∫ eˣ thod simplifies repeated parts: differentiate u column until zero, integrate dv column, then combine with alternating signs.

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有时需要多次运用分部积分，尤其是对于 ∫ x² eˣ dx 或 ∫ eˣ sin x dx 这类积分。对于后者，两次分部后原积分再次出现，可代数求解。表格法可简化多次分部：将 u 逐次求导至零，将 dv 逐次积分，然后交替正负号组合。

7. Integration Using Partial Fractions | 部分分式积分法

ctors are often integrated via partial fractions. For distinct linear factors, write 1/((x–a)(x–b)) = A/(x–a) + B/(x– is essential for integrals arising in differential equations and chemical kinetics models.

分母可分解为一次或二次因式的有理函数常通过部分分式积分。对于不同的一次因式，写成 1/((x–a)(x–b)) = A/(x–a) + B/(x–b)。求出 A 和 B，然后逐项积分得到对数表达式。这对于微分方程和化学动力学模型中的积分至关重要。

8. Trigonometric Integrals – Using Identities | 三角积分 – 使用恒等式

substitution (sin² x + cos² x = 1). If both are even, use half-angle formulas: sin² x = (1 – s² x = (1 + cos 2x)/2. For products of sin(ax) and cos(bx), use product-to-sum identities.

正弦和余弦的幂次积分常需要三角恒等式。对于 ∫ sinᵐ x cosⁿ x dx，若 m 或 n 为奇数，分出一个因式并使用换元（sin² x + cos² x = 1）。若均为偶数，使用半角公式：sin² x = (1 – cos 2x)/2，cos² x = (1 + cos 2x)/2。对于 sin(ax) 与 cos(bx) 的乘积，使用积化和差恒等式。

9. Trigonometric Substitutions for Irrational Integrals | 无理函数的三角代换

When integrands contain √a² – x², √a² + x², or √x² – a², the appropriate trigonometric substitution removes the radical. For √a² – x², set x = a sin θ. For √a² + x², set x = a tan θ. For √x² – a², set x = a sec θ. Then use identities like 1 – sin² θ = cos² θ to simplify. Don’t forget to change dx as well.

当被积函数包含 √a² – x²、√a² + x² 或 √x² – a² 时，恰当的三角代换可消去根号。对 √a² – x²，设 x = a sin θ；对 √a² + x²，设 x = a tan θ；对 √x² – a²，设 x = a sec θ。再利用 1 – sin² θ = cos² θ 等恒等式化简。不要忘记相应地转换 dx。

10. Improper Integrals – Infinite Limits and Discontinuities | 反常积分 – 无穷限与不连续点

o unbounded intervals or integrands with vertical asymptotes. Evaluate ∫ₐ^∞ f(x) dx as lim_b→∞ ntegral converges; otherwise it diverges. For a discontinuity at c in [a,b], split the integral and take one-sided limits. This is vital in probability distributions and physics.

反常积分将概念扩展到无界区间或被积函数具有垂直渐近线的情形。计算 ∫ₐ^∞ f(x) dx 作为 lim_b→∞ ∫ₐᵇ f(x) dx。若极限存在，积分收敛；否则发散。对于在 [a,b] 内 c 点的不连续，需拆分积分并取单侧极限。这在概率分布和物理学中至关重要。

11. Applications – Area Between Curves and Volumes | 应用 – 曲线间面积与体积

Definite integration is used to find the area enclosed between two curves: A = ∫ₐᵇ (f(x) – outer and inner radii. These techniques appear in IB exam Papers and internal assessments.

定积分用于求两曲线间围成的面积：A = ∫ₐᵇ (f(x) – g(x)) dx，其中 f(x) ≥ g(x)。对于旋转体体积，圆盘法：绕 x 轴旋转时 V = π ∫ₐᵇ [R(x)]² dx。垫圈法使用外半径和内半径。这些技巧在IB考试和内部评估中都会出现。

12. Numerical Integration – When Antiderivatives Are Impossible | 数值积分 – 当原函数无法求出时

Not all functions have elementary antiderivatives, e.g., e^–x² h/2 [f(x₀) + 2f(x₁) + 2f(x₂) + ... + f(xₙ)]. Simpson’s rule (n even) gives higher accuracy. IB technology features can compute these approximations.

并非所有函数都有初等原函数，例如 e^–x²。梯形法则等数值方法可近似计算定积分。将 [a,b] 等分为 n 个子区间，宽度为 h，则 Tₙ = h/2 [f(x₀) + 2f(x₁) + 2f(x₂) + ... + f(xₙ)]。辛普森法则（n 为偶数）精度更高。IB技术功能可计算这些近似值。

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