Edexcel Mathematics: Integration Key Points | Edexcel 数学：积分考点精讲

📚 Edexcel Mathematics: Integration Key Points | Edexcel 数学：积分考点精讲

Integration is one of the two central pillars of calculus in the Edexcel A Level Mathematics syllabus. It appears across Pure Mathematics and is essential for solving problems involving areas, volumes, and differential equations. This article distils the must-know facts, standard techniques, and common pitfalls to help you master integration for your Edexcel exams.

积分是 Edexcel A Level 数学大纲中微积分的两大核心支柱之一。它贯穿纯数学部分，对于求解面积、体积和微分方程等问题至关重要。本文提炼了必须掌握的知识点、标准技巧和常见易错点，帮助你攻克 Edexcel 考试中的积分题目。

1. Integration as Reverse Differentiation | 积分作为微分的逆运算

Integration reverses the process of differentiation. If you know the derivative of a function, you can recover the original function up to an additive constant. This is the foundation of indefinite integration.

积分是微分的逆过程。如果你知道一个函数的导数，就能还原出原函数，但会多出一个任意常数。这是不定积分的基础。

For any term xⁿ, where n ≠ −1, the integral follows the power rule: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C. Remember to add the constant of integration C every time you perform an indefinite integral.

对于任何 xⁿ 项（n ≠ −1），积分遵循幂法则：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C。每次求不定积分时，务必加上积分常数 C。

The integral of x⁻¹ is the natural logarithm: ∫ 1/x dx = ln|x| + C. The absolute value ensures the expression is valid for negative x.

x⁻¹ 的积分是自然对数：∫ 1/x dx = ln|x| + C。绝对值确保该表达式对负数 x 也成立。

Standard results for trigonometric, exponential, and logarithmic functions must be memorised. The integral of eˣ is eˣ, and the integral of cos ax is (1/a) sin ax. Always divide by the coefficient of x when integrating in the form f(ax+b).

必须熟记三角函数、指数函数和对数函数的标准积分结果。eˣ 的积分是 eˣ，cos ax 的积分是 (1/a) sin ax。在积分 f(ax+b) 形式时，要始终除以 x 的系数。

2. Definite Integrals and the Fundamental Theorem of Calculus | 定积分与微积分基本定理

A definite integral computes the net area between a curve and the x-axis over a closed interval [a, b]. It is evaluated using the Fundamental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is an antiderivative of f.

定积分计算的是闭区间 [a, b] 上曲线与 x 轴之间的净值面积。它使用微积分基本定理求值：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F 是 f 的一个原函数。

When evaluating, first find the indefinite integral (without +C), then substitute the upper and lower limits and subtract. The constant cancels out, so you never need +C in a definite integral.

计算时，先求出不定积分（不带 +C），然后代入上下限并相减。常数会抵消，因此定积分中不需要写 +C。

If the interval of integration crosses a point where the function changes sign, you must split the integral to find the total area, not the signed area. Many exam questions specifically ask for ‘the area enclosed’, requiring careful handling.

如果积分区间跨越了函数变号的点，则必须拆分积分来求总面积，而非带正负号的面积。很多考题会明确要求求“围成的面积”，需要谨慎处理。

Use a calculator to check your definite integrals when permitted, but always show the full analytical method to earn method marks.

如果允许使用计算器，可以用它检查定积分结果，但务必展示完整的解析步骤才能拿到方法分。

3. Standard Integration Formulae for Edexcel | Edexcel 考试的标准积分公式

The Edexcel formula booklet provides a selection of integrals, but you are expected to know many off by heart. Key results include ∫ sin x dx = -cos x + C and ∫ sec² x dx = tan x + C.

Edexcel 的公式手册提供了一些积分公式，但仍有大量公式需要你牢记。关键结果包括 ∫ sin x dx = -cos x + C 以及 ∫ sec² x dx = tan x + C。

For exponentials: ∫ eᵏˣ dx = (1/k) eᵏˣ + C. For rational functions, remember ∫ 1/(x±a) dx = ln|x±a| + C, and more generally ∫ f'(x)/f(x) dx = ln|f(x)| + C.

指数函数：∫ eᵏˣ dx = (1/k) eᵏˣ + C。对于有理函数，记住 ∫ 1/(x±a) dx = ln|x±a| + C，更一般地，∫ f'(x)/f(x) dx = ln|f(x)| + C。

Trigonometric identities are often needed to rewrite integrands before integrating. For example, use cos 2x = 1 − 2 sin² x to integrate sin² x, or sin 2x = 2 sin x cos x for products.

通常需要先运用三角恒等式将被积函数变形再积分。例如，用 cos 2x = 1 − 2 sin² x 来积分 sin² x，或用 sin 2x = 2 sin x cos x 处理乘积。

For 1/(a² − x²) or 1/(a² + x²), the results are inverse trigonometric or logarithmic functions. These are in the formula booklet, but recognising when to use them is a key skill.

对于 1/(a² − x²) 或 1/(a² + x²)，积分结果是反三角函数或对数函数。这些公式在公式手册里有，但关键技能是识别何时使用它们。

4. Integration by Substitution | 换元积分法

Substitution is used to simplify an integral by changing the variable from x to u. Choose u = g(x) such that the derivative du/dx appears somewhere in the integrand, or can be manipulated to do so.

换元法通过将变量从 x 换为 u 来简化积分。选取 u = g(x) 使得被积函数中出现导数 du/dx，或可通过变形使其出现。

Always rewrite the whole integral in terms of u and du. If the substitution is given in the exam, you must use it. If not, clearly state your choice of u.

始终将整个积分用 u 和 du 重写。如果考题给出了换元式，则必须使用它。如果没有给定，请清楚地写出你选择的 u。

For definite integrals using substitution, you have two options: change the limits to u and evaluate fully in u, or keep the old limits and substitute x back after integrating. Changing limits is usually faster.

对于使用换元法的定积分，有两种方法：将积分限换成 u 并在 u 下完整计算，或保留旧的积分限，积分后再将 u 换回 x。换积分限通常更快捷。

Common substitutions for Edexcel are u = √(x), u = ln x, u = eˣ, and u = sin x or cos x when their derivative partner is present.

Edexcel 考试中常见的换元有 u = √(x)、u = ln x、u = eˣ，以及当另一三角函数出现在导数中时令 u = sin x 或 cos x。

5. Integration by Parts | 分部积分法

Integration by parts comes from the product rule for differentiation. The formula is ∫ u dv = uv − ∫ v du. In simpler notation: ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx.

分部积分法源自微分的乘法法则。公式为 ∫ u dv = uv − ∫ v du。更简洁的写法是：∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx。

The trick is selecting u and dv/dx appropriately. A common mnemonic is LIATE: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential – choose u based on this priority. For Edexcel, ∫ x eˣ dx and ∫ ln x dx are classic examples.

技巧在于恰当地选取 u 和 dv/dx。常用口诀是 LIATE：对数函数、反三角函数、代数函数、三角函数、指数函数——按此优先级选择 u。Edexcel 考试中，∫ x eˣ dx 和 ∫ ln x dx 是经典例子。

Sometimes you need to apply integration by parts twice, especially with products like ∫ x² sin x dx or ∫ eˣ sin x dx. In the latter case, after two applications the original integral reappears, allowing you to solve algebraically.

有时需要两次运用分部积分法，特别对于像 ∫ x² sin x dx 或 ∫ eˣ sin x dx 这样的乘积。后一种情况，两次分部后原积分会再次出现，从而通过代数方法求解。

When using integration by parts, always identify u, du/dx, v, and dv/dx clearly. Marks are awarded for stating these correctly even if the subsequent algebra contains errors.

使用分部积分法时，务必清晰地标出 u、du/dx、v 和 dv/dx。即使后续代数运算有误，正确写出这些仍能得分。

6. Integrating Parametric Equations | 参数方程的积分

For curves defined parametrically as x = f(t), y = g(t), the area under the curve is given by ∫ y dx = ∫ g(t) f'(t) dt, where the limits of t correspond to the required x-values.

对于由参数方程 x = f(t), y = g(t) 定义的曲线，曲线下的面积由 ∫ y dx = ∫ g(t) f'(t) dt 给出，其中 t 的积分限对应所需的 x 值。

This technique appears frequently in Edexcel Pure Mathematics P2 and P3 papers. You must be comfortable with differentiating parametric forms to find dx/dt before setting up the integral.

此技巧频繁出现在 Edexcel 纯数学 P2 和 P3 试卷中。你必须熟练掌握对参数方程求导以找到 dx/dt，然后才能建立积分。

If the curve is given by a set of parametric equations, the volume of revolution about the x-axis is V = π ∫ y² dx = π ∫ [y(t)]² (dx/dt) dt. Similar logic applies for rotation about the y-axis using x² dy.

如果曲线由一组参数方程给出，绕 x 轴的旋转体体积为 V = π ∫ y² dx = π ∫ [y(t)]² (dx/dt) dt。绕 y 轴旋转同理，使用 x² dy。

Always draw a sketch in your working to confirm the direction of integration and the limits. Many mistakes arise from using the wrong t-limits.

解题时最好画出示意图，确认积分方向和积分限。很多错误来源于 t 的积分限用错。

7. Area Bounded by Curves | 曲线围成的面积

To find the area between two curves y = f(x) and y = g(x) from x = a to x = b, use ∫ₐᵇ (top curve − bottom curve) dx. If the curves cross, split the interval.

要求两条曲线 y = f(x) 与 y = g(x) 在 x = a 至 x = b 之间围成的面积，使用 ∫ₐᵇ (上曲线 – 下曲线) dx。如果曲线相交，必须拆分区间。

Sometimes it is more convenient to integrate with respect to y, especially when the curves are given as x = f(y). The area is then ∫ (right curve − left curve) dy with appropriate y-limits.

有时对 y 积分更方便，尤其是当曲线以 x = f(y) 的形式给出时。此时面积为 ∫ (右曲线 – 左曲线) dy，并配上合适的 y 积分限。

In a closed region bounded by more than two curves, identify the points of intersection and set up a series of definite integrals. Clear labelling and annotation help avoid sign errors.

在由两条以上曲线围成的封闭区域内，要找出所有交点并建立一系列定积分。清晰的标注和注释有助于避免符号错误。

Edexcel often asks for the exact value of the area, meaning the final answer should be left in terms of π, e, or surds, not a decimal approximation.

Edexcel 考试常要求面积的精确值，这意味着最终答案应以 π、e 或根式等形式表示，而非小数近似值。

8. Volumes of Revolution | 旋转体体积

When a curve y = f(x) is rotated 360° about the x-axis between x = a and x = b, the volume generated is V = π ∫ₐᵇ [f(x)]² dx. The π factor is the cross-sectional area of a circular disc.

当曲线 y = f(x) 在 x = a 与 x = b 之间绕 x 轴旋转 360° 时，生成的体积为 V = π ∫ₐᵇ [f(x)]² dx。其中 π 因子是圆盘的截面积。

For rotation about the y-axis, the formula becomes V = π ∫ [g(y)]² dy where x = g(y) describes the curve and the limits are y-values.

对于绕 y 轴旋转，公式变为 V = π ∫ [g(y)]² dy，其中曲线描述为 x = g(y)，积分限为 y 值。

If the region between two curves is rotated, the volume is V = π ∫ (outer² − inner²) dx. Be careful to identify which curve is further from the axis of rotation.

如果旋转的是两条曲线之间的区域，则体积为 V = π ∫ (外半径² – 内半径²) dx。要仔细判断哪条曲线离旋转轴更远。

Parametric volumes involve substituting for y and dx in terms of the parameter, as mentioned earlier. Remember to change the limits to the parameter values.

参数方程旋转体体积涉及将 y 和 dx 用参数表示，如前所述。切记将积分限换成参数值。

9. Solving Differential Equations by Integration | 用积分求解微分方程

First-order separable differential equations are solved by rearranging to the form ∫ g(y) dy = ∫ f(x) dx. After integrating both sides, always add a constant of integration on one side.

一阶可分离变量的微分方程通过变形为 ∫ g(y) dy = ∫ f(x) dx 来求解。两边积分后，务必在一边加上积分常数。

Use the initial conditions provided to find the particular solution. If no conditions are given, leave your answer as a general solution containing an arbitrary constant.

利用题目给出的初始条件来求特解。如果没有给出条件，则将答案写成含有任意常数的通解。

Many exam questions embed a real-world context such as temperature change, population growth, or radioactive decay. The differential equation models the rate of change, and integration yields the quantity as a function of time.

许多考题会嵌入实际背景，如温度变化、种群增长或放射性衰变。微分方程描述了变化率，积分后得到关于时间的函数。

When integrating 1/y dy, the result is ln|y|. In applied contexts where y > 0, the modulus is often dropped, but state this explicitly to be safe.

积分 1/y dy 得到 ln|y|。在应用情境下若 y > 0，通常省略绝对值符号，但为了安全起见，应明确说明这一点。

10. Integration by Partial Fractions | 利用部分分式积分

Rational functions with a factorisable denominator can often be expressed as partial fractions, making integration straightforward. The key forms are A/(x±p) + B/(x±q) and repeated factors like A/(x+p) + B/(x+p)².

分母可因式分解的有理函数通常可表为部分分式，从而使积分变得简单。关键形式有 A/(x±p) + B/(x±q)，以及重复因子如 A/(x+p) + B/(x+p)²。

After decomposition, each term integrates to a natural logarithm or a power of x. For example, ∫ A/(x+a) dx = A ln|x+a|, and ∫ A/(x+a)² dx = −A/(x+a).

分解后，每一项的积分变成自然对数或 x 的幂。例如，∫ A/(x+a) dx = A ln|x+a|，而 ∫ A/(x+a)² dx = −A/(x+a)。

An improper rational function (degree of numerator ≥ degree of denominator) must first be simplified by polynomial division before applying partial fractions.

假分式（分子次数 ≥ 分母次数）在应用部分分式前，必须先通过多项式除法进行化简。

This technique ties together algebraic manipulation and integration, and is a highly predictable Edexcel question worth mastering thoroughly.

这套技巧将代数运算与积分结合在一起，是一类非常可预测的 Edexcel 题型，值得扎实掌握。

11. Integration Using Trigonometric Identities | 利用三角恒等式积分

Products and powers of trigonometric functions often cannot be integrated directly but can be simplified using identities. For sin² x and cos² x, use the double-angle formulas: sin² x = ½(1 − cos 2x), cos² x = ½(1 + cos 2x).

三角函数的乘积和幂往往不能直接积分，但可以通过恒等式化简。对于 sin² x 和 cos² x，使用倍角公式：sin² x = ½(1 − cos 2x)，cos² x = ½(1 + cos 2x)。

For odd powers of sin x or cos x, separate one factor and replace the rest using Pythagoras’ identity sin² x + cos² x = 1, then use substitution u = cos x or u = sin x.

对于 sin x 或 cos x 的奇数次幂，分离出一个因子并用 sin² x + cos² x = 1 替换其余部分，然后换元 u = cos x 或 u = sin x。

Integrating tan² x or cot² x is done by rewriting tan² x = sec² x − 1, which gives an easy integral. Similar approach works for cosec² x and sec² x.

积分 tan² x 或 cot² x 时，将其改写为 tan² x = sec² x − 1，积分就变得简单。类似方法适用于 cosec² x 和 sec² x。

Be prepared to simplify integrands like sin 3x cos 2x using product-to-sum formulas, though these are less common in Edexcel than the power-reduction identities.

需准备好使用积化和差公式化简像 sin 3x cos 2x 这样的被积函数，但在 Edexcel 中，这类题不如降幂恒等式那么常见。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

Forgetting the constant of integration in an indefinite integral is one of the most frequent errors. It can cost you a mark even if everything else is correct.

在不定积分中忘记加积分常数是最常见的错误之一。即使其他步骤全对，没有 C 也会失分。

Misidentifying the sign when integrating trigonometric functions, e.g., writing ∫ sin x dx = cos x instead of −cos x. Create a quick mental check: differentiate your answer to see if you get the original.

积分三角函数时符号搞错，例如将 ∫ sin x dx 写成 cos x 而不是 −cos x。可以进行快速心算检查：将答案微分，看是否得到原函数。

In area and volume problems, not checking whether the curve crosses the axis within the interval leads to incorrect net area calculations. Always find x-intercepts or intercepts with the other curve.

在面积和体积问题中，若不检查曲线在区间内是否与坐标轴或其他曲线相交，会导致净值面积计算错误。一定要先求出所有交点。

When using substitution for definite integrals, failing to change the limits to the new variable is a classic slip. Write the new limits beside the original ones to track the change.

在定积分换元时，未将积分限换为新变量的值是一个经典失误。将新积分限写在原积分限旁边，以便跟踪变化。

Read the question carefully: ‘find the area’ often means positive area regardless of sign, whereas ‘evaluate the integral’ means signed area. Highlight keywords to avoid misinterpretation.

仔细读题：“求面积”通常指不考虑符号的正面积，而“计算积分”则指带符号的面积。用高亮笔标出关键词，避免曲解题意。

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Edexcel Mathematics: Integration Key Points | Edexcel 数学：积分 考点精讲