pdfjoiner_(4)-241 | 积分技巧综合训练 (PDF合并4-241)

📚 pdfjoiner_(4)-241 | 积分技巧综合训练 (PDF合并4-241)

This article is designed to consolidate and practise the core integration techniques required for Edexcel A-Level Mathematics, based on the exercises compiled in the pdfjoiner worksheet (4)-241. You will be guided through standard integrals, substitution, parts, partial fractions, trigonometric integration, definite integrals, differential equations, parametric integration, volumes of revolution, and exponential/logarithmic integration. Mastering these methods is essential for success in Pure Mathematics papers.

本文旨在结合 pdfjoiner 练习册 (4)-241 的内容，巩固和训练 Edexcel A-Level 数学所要求的核心积分技巧。我们将逐一梳理标准积分、换元法、分部积分法、部分分式积分、三角积分、定积分、微分方程、参数积分、旋转体体积以及指数与对数函数的积分。掌握这些方法是在纯数学试卷中取得成功的关键。

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1. Standard Integrals and Basic Rules | 标准积分与基本法则

Every integration technique builds on the standard integrals you must memorise. For any constant n ≠ −1, ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. The integral of 1/x is ln|x| + C. For exponentials, ∫ eˣ dx = eˣ + C, and ∫ aˣ dx = aˣ the reverse of differentiation, so you can always check your answer by differentiating it.

每一种积分技巧都建立在你必须熟记的标准积分之上。对于任意常数 n ≠ −1，有 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。1/x 的积分是 ln|x| + C。指数函数方面，∫ eˣ dx = eˣ + C，∫ aˣ dx = aˣ/ln a + C。三角函数的标准结果包括 ∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C 以及 ∫ sec²x dx = tan x + C。请记住，积分是微分的逆运算，因此你可以通过对答案求导来检验结果。

∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ −1

Function	Integral
xn (n≠−1)	xn+1/(n+1) + C
1/x	ln|x| + C
ex	ex + C
sin x	−cos x + C
cos x	sin x + C

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2. Integration by Substitution | 换元积分法

on is used when an integral contains a composite function, typically of the form f(g(x))·g' ), then find du/dx and rewrite the integral in terms of u and du. After integration, substitute back to the original variable. For definite integrals, you can either change the limits to u-values or revert to x before evaluating. Always look for a function and its derivative appearing together.

当被积函数为复合函数形式，通常形如 f(g(x))·g'(x) 时，使用换元法。令 u = g(x)，求出 du/dx，并将积分完全用 u 和 du 表示。积分完成后，回代到原变量。对于定积分，既可以将积分限转换为 u 值，也可以在还原为 x 后再代入上下限。务必留意函数与其导数同时出现的情况。

∫ 2x·cos(x²) dx → let u = x², du = 2x dx → ∫ cos u du = sin u + C = sin(x²) + C

• Identify u = inner function
• Compute du/dx and express dx = du / (du/dx)
• Rewrite the integral completely in u
• Integrate, then substitute back

• 确定 u = 内层函数
• 计算 du/dx，并将 dx 表示为 du / (du/dx)
• 将积分完全用 u 写出
• 积分，然后回代

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3. Integration by Parts | 分部积分法

y follows the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) where u is taken from the earlier category. In Edexcel questions, typical examples include ∫ x·eˣ , ∫ x·ln x dx, and ∫ x·sin x dx. Sometimes you need to apply the method twice, as with ∫ eˣ sin x dx.

分部积分法逆转了乘积的微分法则。公式为 ∫ u dv = uv − ∫ v du。需要合理选择 u 和 dv；通常采用 LIATE 法则（对数、反三角、代数、三角、指数），u 优先从靠前的类别中选取。在 Edexcel 试题中，典型例子包括 ∫ x·eˣ dx、∫ x·ln x dx 以及 ∫ x·sin x dx。有时需要连续两次使用分部积分，例如 ∫ eˣ sin x dx。

∫ u dv = uv − ∫ v du

Integral	u choice	Result
∫ x ex dx	x (algebraic)	ex (x−1) + C
∫ ln x dx	ln x (log)	x ln x − x + C

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4. Integration Using Partial Fractions | 利用部分分式积分

When integrating rational functions where the numerator is of lower degree than the denominator, partial fraction decomposition is a powerful tool. Factorise the denominator into linear and irreducible quadratic factors. For each distinct linear factor (ax + b), include a term A/(ax + b). Repeated factors require A/(ax+b) + B/(ax+b)². After finding the constants, the integral becomes a sum of simple logarithmic or arctangent forms. This technique often appears in Edexcel P3 and P4 papers.

对分子次数低于分母的有理函数积分时，部分分式分解是一项强有力的工具。将分母因式分解为一次和不可约二次因式。对于每个不同的一次因式 (ax + b)，包含一项 A/(ax + b)；重复因式则需要 A/(ax+b) + B/(ax+b)²。求出常数后，积分就转化为简单的对数或反正切形式之和。这种技巧经常出现在 Edexcel P3 和 P4 试卷中。

∫ (2x+3)/(x²+x−2) dx = ∫ [1/(x−1) + 1/(x+2)] dx = ln|x−1| + ln|x+2| + C

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5. Trigonometric Integrals and Identities | 三角积分与恒等式

os²x = ½(1 + cos 2x), and sin 2x = 2 sin x cos x. For products like sin³x or cos³x, save one factor and convert the rest using the Pythagorean identity. Integrals of tan x and cot x require rewriting as sin/cos and using substitution or recognising ln|sec x|. The strategy is to reduce powers or convert to manageable forms.

三角函数积分往往需要利用恒等式将被积函数化为标准形式。常用恒等式有 sin²x = ½(1 − cos 2x)、cos²x = ½(1 + cos 2x) 以及 sin 2x = 2 sin x cos x。对于 sin³x 或 cos³x 这类乘积，保留一个因式，其余用勾股恒等式转换。tan x 和 cot x 的积分需要写为 sin/cos 形式，然后使用换元法或直接识别为 ln|sec x|。核心策略是降次或转换为易于处理的形式。

∫ sin²x dx = ∫ ½(1 − cos 2x) dx = ½(x − ½ sin 2x) + C

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6. Definite Integration and Area Problems | 定积分与面积问题

e; those below are negative. To find the total enclosed area, you must split the interval where the function crosses the axis and take absolute values. Area between two curves is found by integrating the difference f(x) − g(x). Always sketch the region first to avoid sign errors.

定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 之间的带号面积。轴上方面积为正，下方面积为负。求包围的总面积时，需要在函数穿越 x 轴的位置拆分区间并取绝对值。两条曲线之间的面积通过积分差值 f(x) − g(x) 来求得。一定要先画出草图以避免符号错误。

Area = ∫ₐᵇ (top curve − bottom curve) dx

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7. Differential Equations: Separation of Variables | 微分方程：分离变量法

by separating variables: rewrite as (1/g(y)) dy = f(x) dx, then integrate both sides. Don' tion growth, cooling, and chemical reaction models. Be careful to express the solution as y = ... if possible, and simplify the constant.

形如 dy/dx = f(x)g(y) 的一阶微分方程可以通过分离变量法求解：改写为 (1/g(y)) dy = f(x) dx，然后对两边积分。切勿遗漏积分常数，通常通过初始条件确定其值。在 Edexcel A-Level 中，该方法应用于人口增长、冷却和化学反应模型。尽可能将解表示为 y = ... 的形式，并化简常数。

dy/dx = 2xy → ∫ 1/y dy = ∫ 2x dx → ln|y| = x² + C → y = Ae^x²

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8. Parametric Integration | 参数积分

lues. The arc length and surface area are beyond A-Level, but you must be comfortable finding areas. Always rewrite y and dx using dt. This method is particularly useful for curves that cannot be expressed as y = f(x) easily, such as cycloids or certain Cartesian loops.

当曲线由参数方程 x = x(t)、y = y(t) 定义时，曲线下的面积由 ∫ y (dx/dt) dt 给出，t 的上下限与 x 值相对应。弧长和表面积超出了 A-Level 范围，但必须熟练掌握求面积。始终利用 dt 重写 y 和 dx。对于难以表示为 y = f(x) 的曲线（如摆线或某些笛卡尔环），这种方法尤为有效。

Area = ∫ y(t) · (dx/dt) dt (with t-limits)

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9. Volumes of Revolution | 旋转体体积

axis, use V = π ∫ₐᵇ x² dy, where x is expressed in terms of y. For parametric curves, adapt the formula to integrate with respect to the parameter. Volumes formed between two curves are found by subtracting the inner volume from the outer. This topic is a favourite in the Edexcel Pure Mathematics 2/3/4 sections.

旋转体是由一条曲线绕 x 轴或 y 轴旋转所生成的立体。绕 x 轴旋转的体积公式为 V = π ∫ₐᵇ y² dx；绕 y 轴则使用 V = π ∫ₐᵇ x² dy，其中 x 用 y 表示。对于参数曲线，需调整公式以对参数积分。两条曲线之间形成的旋转体体积通过外体积减去内体积求得。这一专题是 Edexcel 纯数学 P2/P3/P4 部分的热门考点。

V = π ∫ y² dx (rotation about x-axis)

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10. Integrating Exponential and Logarithmic Functions | 指数与对数函数的积分

Exponential integrals are straightforward: ∫ e^kx dx = (1/k) e^kx + C. When the integrand is of the form f' hms, the standard integral ∫ ln x dx = x ln x − x + C is derived via parts. Integrals of aˣ are handled using aˣ = e^{x ln a}. In applied contexts, eˣ models appear in growth/decay problems; linking integration back to differential equations is a crucial skill.

指数函数的积分十分直接：∫ e^kx dx = (1/k) e^kx + C。当被积函数形如 f'(x) e^(f(x)) 时，结果为 e^(f(x)) + C。对于对数函数，标准积分 ∫ ln x dx = x ln x − x + C 系通过分部积分推出。aˣ 类积分可借助 aˣ = e^{x ln a} 处理。在应用题中，eˣ 模型常出现在增长/衰减问题中；将积分与微分方程相关联是一项关键技能。

• ∫ e^2x dx = ½ e^2x + C
• ∫ x e^x² dx = ½ e^x² + C (by substitution)
• ∫ ln(3x) dx = x ln(3x) − x + C

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