📚 GCSE WJEC Maths: Integration Essentials | GCSE WJEC 数学：积分 考点精讲
Integration is a fundamental tool in mathematics that allows us to reverse differentiation. In the WJEC GCSE curriculum, you will learn how integration helps us find original functions, compute areas under curves, and solve real‑world problems involving motion. This article breaks down every key concept you need to master for the exam.
积分是数学中的一个基本工具，它让我们能够逆转求导过程。在 WJEC 的 GCSE 课程中，你将学习如何通过积分找到原函数、计算曲线下的面积，并解决涉及运动的实际问题。这篇文章将逐一拆解你在考试中必须掌握的每一个核心概念。
1. What is Integration? | 什么是积分？
Integration is the process of finding a function when its derivative is given. In other words, it is the reverse operation of differentiation. If differentiating takes us from a function to its gradient function, integrating takes us back from the gradient function to the original function.
积分是已知一个函数的导数，求出该函数的过程。换句话说，它是微分的逆运算。如果把微分看作从原函数到其导函数的变换，那么积分就是由导函数回溯到原函数的方法。
The symbol for integration is an elongated ‘S’: ∫, and the expression to be integrated is called the integrand. The result is called the integral or antiderivative.
积分的符号是一个拉长的 “S”：∫，被积分的表达式称为被积函数。得到的结果称为积分，也叫原函数。
2. Integration as Reverse Differentiation | 积分作为微分的逆运算
When we differentiate a polynomial, we multiply by the power and reduce the power by 1. To reverse this, when integrating, we increase the power by 1 and then divide by the new power.
当我们对多项式求导时，我们用指数乘以系数，并把指数减 1。反向操作时，积分就需要 先把指数加 1，然后再 除以新的指数。
For example, if d/dx (x³) = 3x², then integrating 3x² gives x³. Notice the process: the exponent moved from 3 to 2 in differentiation, so to integrate 3x² we go from exponent 2 to 3.
例如，d/dx (x³) = 3x²，那么对 3x² 积分就会得到 x³。注意这个过程：求导时指数从 3 变为 2，因此积分 3x² 时指数要从 2 回到 3。
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, provided n ≠ −1
3. The Indefinite Integral | 不定积分
An indefinite integral is an integral without limits. It represents a family of functions because any constant term vanishes when differentiated. We write it as ∫ f(x) dx and always add a constant of integration, denoted by ‘c’.
不定积分是没有积分限的积分。它表示一族函数，因为任何常数项在求导后都会消失。我们把它记作 ∫ f(x) dx，并且总是加上一个积分常数，用 ‘c’ 表示。
For instance, integrating 2x gives x² + c. All the functions x² + 1, x² − 5, x² + 100 have the same derivative 2x, so the ‘+ c’ covers every possible constant.
例如，对 2x 积分会得到 x² + c。函数 x² + 1、x² − 5、x² + 100 的导数都是 2x，所以 ‘+ c’ 涵盖了所有可能的常数。
4. Constant of Integration ‘c’ | 积分常数 c
The constant of integration is crucial because integration reverses differentiation, and differentiation loses any constant term. Without ‘c’, we would have only one of infinitely many possible original functions.
积分常数至关重要，因为积分是微分的逆过程，而微分会丢失所有常数信息。如果没有 ‘c’，我们只能得到无数个可能原函数中的一个。
In exam questions, you are often given extra information – such as a point on the curve – to find the specific value of ‘c’. Always remember to write + c and then use the given condition to find its numerical value.
在考试题目中，通常会给出额外信息（比如曲线上的一个点）来求出 ‘c’ 的具体数值。请务必记住先写出 + c，再利用给定条件求出它的数值。
5. Basic Integration Rules | 基本积分法则
The rule ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c works for all real numbers n except n = −1. When integrating a constant k, treat it as k × x⁰, giving ∫ k dx = kx + c.
规则 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c 适用于所有 n ≠ −1 的实数。当对常数 k 积分时，可以看作 k × x⁰，因此 ∫ k dx = kx + c。
We can also integrate sums and differences term by term: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx. Constant multipliers can be taken outside the integral: ∫ a f(x) dx = a ∫ f(x) dx.
我们还可以逐项对和与差进行积分：∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。常数因子可以提到积分号外面：∫ a f(x) dx = a ∫ f(x) dx。
6. Integrating Polynomials | 多项式积分
To integrate a polynomial like 4x³ − 2x² + 5x − 7, apply the power rule to each term separately. For 4x³: increase power to 4, divide by 4 → x⁴. For −2x²: power to 3, divide by 3 → −⅔ x³. For 5x: power to 2, divide by 2 → (5/2)x². For −7: ∫ −7 dx = −7x. Then add + c.
要对多项式 4x³ − 2x² + 5x − 7 进行积分，只需逐项应用幂规则。对 4x³：增加指数到 4，除以 4 → x⁴。对 −2x²：指数到 3，除以 3 → −⅔ x³。对 5x：指数到 2，除以 2 → (5/2)x²。对 −7：∫ −7 dx = −7x。最后加上 + c。
∫ (4x³ − 2x² + 5x − 7) dx = x⁴ − (2/3)x³ + (5/2)x² − 7x + c
7. Finding the Original Function | 求原函数
When you know the derivative f'(x) and a point on the original curve f(x), you can find the equation of f(x). First integrate f'(x) to get f(x) with a ‘+ c’. Then substitute the given x and y values to solve for c.
当你已知导数 f'(x) 以及原曲线 f(x) 上的一点，就可以求出 f(x) 的方程。首先对 f'(x) 积分，得到带 ‘+ c’ 的 f(x)。然后代入所给的 x 和 y 值，解出 c。
Example: f'(x) = 6x² − 4, and the curve passes through (1, 5). Integrate: f(x) = 2x³ − 4x + c. Substitute x=1, y=5: 5 = 2(1)³ − 4(1) + c → 5 = 2 − 4 + c → c = 7. So f(x) = 2x³ − 4x + 7.
例题：已知 f'(x) = 6x² − 4，且曲线经过点 (1, 5)。积分得：f(x) = 2x³ − 4x + c。代入 x=1, y=5：5 = 2(1)³ − 4(1) + c → 5 = 2 − 4 + c → c = 7。因此 f(x) = 2x³ − 4x + 7。
8. Definite Integration | 定积分
A definite integral has limits of integration – a lower limit a and an upper limit b – written as ∫ₐᵇ f(x) dx. It gives a numerical value, not a family of functions. The constant ‘c’ is not needed because it cancels out when evaluating the difference.
定积分带有积分限——下限 a 和上限 b，写作 ∫ₐᵇ f(x) dx。它给出一个数值，而不是一族函数。此时不需要积分常数 c，因为在计算差值时会相互抵消。
To evaluate, find any antiderivative F(x) of f(x), then compute F(b) − F(a). This is the Fundamental Theorem of Calculus applied at GCSE level.
要计算定积分，先找到 f(x) 的任意一个原函数 F(x)，然后计算 F(b) − F(a)。这就是 GCSE 阶段用到的基本微积分定理。
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)
9. Area Under a Curve | 曲线下的面积
The definite integral calculates the exact area between a curve y = f(x), the x‑axis, and the vertical lines x = a and x = b, provided the curve lies above the x‑axis on that interval. If the curve goes below the x‑axis, the integral gives a negative value – always split the area into above‑axis and below‑axis sections.
定积分用于计算曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 之间的精确面积，前提是在该区间内曲线位于 x 轴上方。如果曲线有一部分在 x 轴下方，积分值会为负——一定要将区域分成轴上部分和轴下部分分别处理。
For example, the area between y = x² + 1, x = 0, x = 2 and the x‑axis is found by evaluating ∫₀² (x² + 1) dx = [x³/3 + x]₀² = (8/3 + 2) − 0 = 14/3 square units.
例如，由 y = x² + 1、x = 0、x = 2 和 x 轴围成的面积，可通过计算 ∫₀² (x² + 1) dx 得到：原函数为 [x³/3 + x]₀² = (8/3 + 2) − 0 = 14/3 平方单位。
10. The Trapezium Rule | 梯形法则
When a function is too difficult to integrate exactly, or when we only have a table of values, we can approximate the area under the curve using the Trapezium Rule. This is a numerical method that splits the area into several trapeziums of equal width.
当被积函数难以精确积分，或者我们只有一个数值表时，可以使用梯形法则来近似计算曲线下的面积。这是一种数值方法，将整个区域分割成若干个等宽的梯形。
If the interval [a, b] is divided into n strips of equal width h = (b − a)/n, then the approximate area A ≈ h/2 × [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where yᵢ are the function values at equally spaced x‑coordinates.
如果将区间 [a, b] 分成 n 个等宽的条形，每个宽度 h = (b − a)/n，那么近似面积 A ≈ h/2 × [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 yᵢ 是等距 x 坐标上的函数值。
This method is particularly useful in GCSE questions where you are given a speed‑time graph and asked to estimate the total distance travelled.
当题目给出速度‑时间图并要求估算总行驶距离时，这种方法在 GCSE 题目中尤其常用。
11. Interpreting Gradients and Areas in Kinematics | 运动学中的梯度和面积解释
In a velocity‑time graph, the gradient at any point gives the acceleration, while the area under the graph between two times gives the displacement (total distance if motion is in a straight line and direction doesn’t change). Integration is the mathematical tool that connects these ideas.
在速度‑时间图中，任意一点的梯度表示加速度，而两个时刻之间 图线下的面积 表示位移（如果运动沿直线且方向不变，就是总路程）。积分就是将这两个概念联系起来的数学工具。
If velocity can be expressed as a polynomial function of time, integrating it gives the displacement function. The constant of integration then corresponds to the initial displacement.
如果速度可以表达为时间的多项式函数，对其进行积分就可以得到位移函数。此时的积分常数就对应初始位移。
12. Common Mistakes | 常见错误
Forgetting to add the constant of integration in indefinite integrals is the most frequent error. Another common slip is dividing by the old power instead of the new power – remember the power increases by 1, then divide by that increased power.
最常见错误是在不定积分中忘记添加积分常数。另一个常见失误是用旧的指数去除，而不是用新的指数——记住指数要先加 1，再除以这个加了 1 后的指数。
Students also sometimes integrate a product by integrating each factor separately, which is incorrect. There is no simple product rule for integration; you must always simplify or expand first if possible. Finally, check that your antiderivative differentiates back to the original function – this is a powerful way to verify your answer.
学生有时也会分别对乘积的每个因子积分，这是错误的。积分没有简单的乘法规则；一定要先化简或展开。最后，检查你的原函数求导后是否能还原为被积函数——这是验证答案的绝佳方法。
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