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

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

Integration lies at the heart of calculus, reversing differentiation to solve problems involving accumulation, area and motion. In both IB and Edexcel specifications, mastering integration techniques is essential for tackling everything from simple antiderivatives to challenging differential equations. This revision guide distils the key concepts you need, with clear bilingual explanations and practical examples.

积分是微积分的核心，通过逆转微分来解决涉及累积量、面积和运动的问题。在IB和Edexcel的考试大纲中，掌握积分技巧对于处理从简单的反导数到复杂的微分方程等各类问题至关重要。这份复习指南浓缩了你需要掌握的关键概念，并提供清晰的双语解释和实用示例。

1. What Is Integration? | 什么是积分？

Integration is the process of recovering a function from its given rate of change. If we know the derivative of a function, the integral can help us find the original function, often called an antiderivative.

积分是从已知的变化率恢复原函数的过程。如果知道一个函数的导数，积分就可以帮助我们找到原函数，通常称为反导数。

In notation, the indefinite integral of a function f(x) with respect to x is written as ∫ f(x) dx. The symbol ∫ is an elongated S, representing summation.

用符号表示，函数 f(x) 关于 x 的不定积分记作 ∫ f(x) dx。符号 ∫ 是一个拉长的 S，代表求和。

The result always includes an arbitrary constant C, because the derivative of any constant is zero. Thus, ∫ f(x) dx = F(x) + C, where F'(x) = f(x).

结果总是包含任意常数 C，因为任何常数的导数为零。因此，∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。

d/dx (x²) = 2x ⇔ ∫ 2x dx = x² + C

This inverse relationship between differentiation and integration is the cornerstone of calculus and is used extensively in both IB Analysis & Approaches and Edexcel Pure Mathematics.

这种微分与积分之间的互逆关系是微积分的基石，在IB分析与方法和Edexcel纯数学中都有广泛应用。

2. Indefinite Integrals and Antiderivatives | 不定积分与反导数

An indefinite integral represents a family of functions that differ only by a constant. For a continuous function f(x), if F is any function satisfying F'(x) = f(x), then all antiderivatives are given by F(x) + C.

不定积分表示一族仅相差一个常数的函数。对于连续函数 f(x)，如果存在函数 F 满足 F'(x) = f(x)，那么所有反导数都可以表示为 F(x) + C。

Finding an antiderivative is often a matter of reversing basic differentiation rules, but a solid grasp of standard forms is essential for speed and accuracy in exams.

求反导数往往是逆转基本微分法则的过程，但牢固掌握标准形式对于考试中的速度和准确性至关重要。

The table below summarises the most common indefinite integrals you must memorise:

下表总结了你必须牢记的最常见的不定积分：

Function f(x)	Indefinite Integral ∫ f(x) dx
xⁿ (n ≠ -1)	xⁿ⁺¹ / (n+1) + C
1/x	ln \|x\| + C
eˣ	eˣ + C
aˣ (a > 0)	aˣ / ln a + C
sin x	-cos x + C
cos x	sin x + C
sec² x	tan x + C
1/√(1 – x²)	arcsin x + C
1/(1 + x²)	arctan x + C

These forms are the building blocks for more complicated integrals and appear across both IB and Edexcel papers, sometimes in composite forms requiring substitution.

这些形式是处理更复杂积分的基础，并出现在IB和Edexcel的试卷中，有时会以需要换元法的复合形式出现。

3. Basic Integration Rules | 基本积分法则

Integration obeys linearity rules that simplify calculations. The constant multiple rule states that ∫ k·f(x) dx = k ∫ f(x) dx, and the sum rule allows term-by-term integration: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.

积分遵循可简化计算的线性性质。常数倍法则表明 ∫ k·f(x) dx = k ∫ f(x) dx，加法法则允许逐项积分：∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。

In IB and Edexcel exams, these rules are frequently used when integrating polynomial expressions or splitting rational functions into simpler parts.

在IB和Edexcel考试中，这些法则广泛应用于积分多项式，或将有理函数拆分为更简单的部分。

The power rule for integration is a direct consequence: for any real number n ≠ -1, ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. Special care is needed when n = -1, leading to the natural logarithm.

幂函数的积分法则是直接推论：对于任意实数 n ≠ -1，∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。当 n = -1 时需要特别处理，此时结果为自然对数。

Always simplify the integrand algebraically before integrating. For example, rewrite √x as x½ and 1/x² as x⁻² to use the power rule directly.

在进行积分之前，务必通过代数方法简化被积函数。例如，将 √x 改写为 x½，将 1/x² 改写为 x⁻²，从而直接应用幂法则。

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

A definite integral calculates the net signed area between the curve y = f(x) and the x-axis over the interval [a, b]. It is denoted by ∫ from a to b f(x) dx.

定积分计算曲线 y = f(x) 与 x 轴在区间 [a, b] 上的净有号面积，记作 ∫ from a to b f(x) dx。

The Fundamental Theorem of Calculus (FTC) links differentiation and integration: if F'(x) = f(x) and f is continuous on [a, b], then ∫ from a to b f(x) dx = F(b) – F(a).

微积分基本定理（FTC）将微分与积分联系起来：如果 F'(x) = f(x) 且 f 在 [a, b] 上连续，那么 ∫ from a to b f(x) dx = F(b) – F(a)。

∫ from a to b f(x) dx = [F(x)]ₐᵇ = F(b) – F(a)

This theorem transforms the area problem into finding an antiderivative, which is straightforward once the techniques in Section 2 are mastered.

该定理将面积问题转化为寻找反导数，一旦掌握了第2节中的技巧，这个问题就变得非常简单。

Definite integrals can be evaluated even if the curve crosses the x-axis, but interpretation of area requires splitting the interval to avoid sign cancellation.

即使曲线穿过 x 轴，也可以计算定积分，但解释面积时需要拆分区间以避免正负相互抵消。

5. Area Under a Curve | 曲线下方面积

The area between a curve y = f(x) and the x-axis from x = a to x = b is given by ∫ from a to b |f(x)| dx if f changes sign. In practice, we identify x-intercepts and compute the sum of absolute values of the definite integrals over each subinterval.

曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间的面积，如果 f 变号，则需用 ∫ from a to b |f(x)| dx 计算。实际操作中，我们可以找出 x 轴截点，并对每个子区间上的定积分绝对值求和。

For regions bounded by two curves, say f(x) and g(x) where f(x) ≥ g(x) on [a, b], the area is ∫ from a to b [f(x) – g(x)] dx. This is a classic IB and Edexcel exam question.

对于由两条曲线围成的区域，设 f(x) 和 g(x) 在 [a, b] 上满足 f(x) ≥ g(x)，则面积为 ∫ from a to b [f(x) – g(x)] dx。这是IB和Edexcel考试中的经典题型。

Always sketch the region to avoid mistakes. Visualising the area helps determine the limits of integration and which function is on top.

务必绘制区域草图以避免错误。将区域可视化，有助于确定积分限以及哪条曲线在上方。

When the independent variable is y, the area between a curve x = g(y) and the y-axis from y = c to y = d is ∫ from c to d |g(y)| dy. A similar principle applies.

当自变量为 y 时，曲线 x = g(y) 与 y 轴在 y = c 到 y = d 之间的面积为 ∫ from c to d |g(y)| dy，原理类似。

6. Integration by Substitution | 换元积分法

Integration by substitution is the reverse of the chain rule for differentiation. It is used when an integrand contains a function and its derivative in a nested form.

换元积分法是微分链式法则的逆运算，适用于被积函数以嵌套形式包含一个函数及其导数的情况。

The general method: choose u = g(x), then compute du = g'(x) dx. Rewrite the integral in terms of u and du, integrate, and finally replace u with g(x).

一般步骤：令 u = g(x)，计算 du = g'(x) dx。将积分用 u 和 du 表示，进行积分，最后用 g(x) 替换回 u。

∫ f(g(x)) g'(x) dx = ∫ f(u) du

For example, to find ∫ 2x·sin(x²) dx, let u = x², so du = 2x dx. The integral becomes ∫ sin u du = -cos u + C = -cos(x²) + C.

例如，求 ∫ 2x·sin(x²) dx，令 u = x²，则 du = 2x dx。积分化为 ∫ sin u du = -cos u + C = -cos(x²) + C。

In definite integrals, substitution requires changing the limits. If u = g(x), the lower limit becomes g(a) and the upper limit becomes g(b). This avoids reverting to x.

对于定积分，换元时需要同时变换积分限。如果 u = g(x)，下限变为 g(a)，上限变为 g(b)。这样可以避免再换回 x 的步骤。

Always look for a suitable inner function whose derivative appears as a factor. Common substitutions involve linear functions, exponentials and trigonometric arguments.

务必寻找合适的内部函数，使其导数作为一个因子出现。常见的换元涉及线性函数、指数函数和三角函数的角变量。

7. Integration by Parts | 分部积分法

Integration by parts is derived from the product rule for differentiation. It is used when the integrand is a product of two types of functions, such as polynomial × exponential or polynomial × trigonometric.

分部积分法源于微分的乘法法则，适用于被积函数为两类函数的乘积，例如多项式乘以指数函数，或多项式乘以三角函数。

∫ u dv = u v – ∫ v du

Choose u as the function that simplifies when differentiated, and dv as the rest, including dx. Then compute du by differentiating u, and v by integrating dv.

选择 u 为求导后简化的函数，将剩余部分（包括 dx）设为 dv。然后通过求导得到 du，通过对 dv 积分得到 v。

A helpful mnemonic for choosing u is LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Functions higher on the list are usually better choices for u.

选择 u 的一个实用助记法是 LIATE：对数函数、反三角函数、代数函数、三角函数、指数函数。排列越靠前的函数通常越适合选作 u。

For example, ∫ x·eˣ dx: let u = x, dv = eˣ dx, then du = dx, v = eˣ. Applying the formula gives x eˣ – ∫ eˣ dx = x eˣ – eˣ + C.

例如，∫ x·eˣ dx：令 u = x，dv = eˣ dx，则 du = dx，v = eˣ。应用公式得到 x eˣ – ∫ eˣ dx = x eˣ – eˣ + C。

Repeated integration by parts may be necessary, as in ∫ x² sin x dx, where the method must be applied twice. Tabular integration can streamline such repeated cases.

有时需要反复运用分部积分法，例如 ∫ x² sin x dx 需使用两次。表格积分法可以简化这类多次分部积分的过程。

8. Definite Integrals with Advanced Techniques | 定积分与高级技巧

Both substitution and integration by parts apply to definite integrals. The key difference is that with substitution, limits are adjusted, whereas with parts, the uv term is evaluated at the original limits.

换元积分法和分部积分法都适用于定积分。关键区别在于：使用换元法时需调整积分限，而使用分部积分法时，uv 项在原积分限处取值。

For integration by parts with definite integral: ∫ from a to b u dv = [u v]ₐᵇ – ∫ from a to b v du, where the bracket denotes evaluation at the limits.

对于定积分的分部积分：∫ from a to b u dv = [u v]ₐᵇ – ∫ from a to b v du，其中方括号表示在上下限处求值。

Substitution in definite integrals simplifies evaluation because there is no need to convert back to x. Just remember to change the limits exactly: if u = g(x), lower limit = g(a), upper = g(b).

在定积分中使用换元法可以简化计算，因为无需换回 x。只需牢记精确变换积分限：若 u = g(x)，下限 = g(a)，上限 = g(b)。

Exam questions often require combining techniques, for instance, using substitution to simplify the integrand and then parts to finish. Always write out the transformed integral clearly to avoid errors.

考试题目常要求综合运用多种技巧，例如先用换元法简化被积函数，再用分部积分法求解。务必清晰地写出变换后的积分，以避免错误。

9. Numerical Integration: The Trapezoidal Rule | 数值积分：梯形法则

When an integral cannot be evaluated analytically, or the function is given only as a table of values, numerical methods like the trapezoidal rule become essential. Both IB and Edexcel include this topic.

当积分无法解析求解，或函数仅以数据表格形式给出时，梯形法则等数值方法就变得至关重要。IB和Edexcel大纲都包含这一主题。

The trapezoidal rule approximates the area under y = f(x) by dividing the interval [a, b] into n equal strips of width h = (b – a)/n, and using trapezoids to fit the curve.

梯形法则通过将区间 [a, b] 等分为 n 个宽度为 h = (b – a)/n 的小条，并用梯形逼近曲线，来近似计算 y = f(x) 下的面积。

∫ from a to b f(x) dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + … + y_{n-1}) + yₙ]

Here y₀ = f(a), y₁ = f(a+h), …, yₙ = f(b). The more strips used, the better the approximation, though the error depends on the curvature of f.

其中 y₀ = f(a)，y₁ = f(a+h)，…，yₙ = f(b)。使用的条带越多，近似效果越好，但误差取决于 f 的曲率。

The formula is easy to apply using a table of ordinates. In both IB and Edexcel, you may need to state whether the estimate is an overestimate or underestimate based on the curve’s concavity.

利用纵坐标表格可以轻松使用该公式。在IB和Edexcel考试中，你可能需要根据曲线的凹凸性判断估值是高估还是低估。

10. Differential Equations and Integration | 微分方程与积分

Integration is the key to solving first-order ordinary differential equations, particularly those of the form dy/dx = f(x)g(y). The method of separation of variables isolates y terms on one side and x terms on the other.

积分是求解一阶常微分方程的关键，尤其是形如 dy/dx = f(x)g(y) 的方程。分离变量法将含有 y 的项移到一边，含有 x 的项移到另一边。

∫ 1/g(y) dy = ∫ f(x) dx

After integration, a general solution appears with an arbitrary constant. An initial condition, e.g. y(x₀) = y₀, allows determination of the particular solution.

积分后，通解含有一个任意常数。利用初始条件，例如 y(x₀) = y₀，可以确定特解。

For example, to solve dy/dx = xy, separate to ∫ 1/y dy = ∫ x dx, giving ln |y| = x²/2 + C, so y = A e^(x²/2) where A = ±e^C.

例如，求解 dy/dx = xy，分离得 ∫ 1/y dy = ∫ x dx，得到 ln |y| = x²/2 + C，因此 y = A e^(x²/2)，其中 A = ±e^C。

Models in kinematics, population growth and cooling are often governed by such equations. Both IB and Edexcel tests ask students to set up, solve and interpret these models.

运动学、人口增长和冷却模型中的问题常由这类方程控制。IB和Edexcel考试都要求学生建立、求解并解释这些模型。

Always present the solution in the form requested, and check whether the answer should be explicit (y = …) or can remain implicit.

务必按照题目要求的形式给出解，并检查答案应该是显式 (y = …) 还是可以保留隐式。

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IB Edexcel Mathematics: Integration Key Concepts | IB Edexcel 数学：积分 考点精讲