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

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

Integration is the reverse process of differentiation. In IGCSE Edexcel Mathematics, you are expected to find indefinite integrals of simple polynomial functions, determine the constant of integration using given conditions, and evaluate definite integrals to calculate the area under a curve or between curves. This article covers all the essential techniques and typical exam-style problems you need to master.

积分是微分的逆运算。在 IGCSE Edexcel 数学中，你需要掌握简单多项式函数的不定积分，利用给定条件确定积分常数，并计算定积分以求曲线下方或曲线之间的面积。本文涵盖所有关键技巧和典型考题，帮助你全面备考。

1. Integration as Reverse Differentiation | 积分：微分的逆运算

If differentiating takes us from y to dy/dx, integration takes us back from dy/dx to y. For example, if d/dx (x²) = 2x, then ∫ 2x dx = x² + c. The ‘+ c’ is the constant of integration, which appears because differentiation removes any constant term.

如果微分将 y 变为 dy/dx，那么积分就是从 dy/dx 回到 y。例如，d/dx (x²) = 2x，则 ∫ 2x dx = x² + c。这里的 “+ c” 是积分常数，因为微分会消去任何常数项，所以积分时需要加回。

2. The Power Rule for Integration | 幂函数积分法则

For any real number n ≠ -1, ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c. This is the fundamental rule you will use most often. Simply add 1 to the power and divide by the new power, then add the constant of integration.

对于任意实数 n ≠ -1，∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c。这是最核心的积分法则：指数加 1，然后除以新指数，最后加上积分常数。

3. Integrating a Constant | 常数的积分

A constant term k can be thought of as k·x⁰. Applying the power rule: ∫ k dx = ∫ k·x⁰ dx = k·(x¹)/1 + c = kx + c. So the integral of a constant is simply the constant multiplied by x, plus c.

常数 k 可以看作 k·x⁰。应用幂法则：∫ k dx = ∫ k·x⁰ dx = k·(x¹)/1 + c = kx + c。因此常数的积分就是常数乘以 x，再加上 c。

4. Integrating Sums and Differences | 和与差的积分

Integration is linear, meaning ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx. You can integrate a polynomial term by term. Also, constant factors can be taken outside the integral: ∫ a·f(x) dx = a·∫ f(x) dx.

积分具有线性性质，即 ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。你可以逐项积分多项式。另外，常数因子可以提到积分号外：∫ a·f(x) dx = a·∫ f(x) dx。

5. Example: Integrating a Polynomial | 示例：多项式的积分

Find ∫ (4x³ – 3x² + 2x – 5) dx. Integrate each term: 4·(x⁴/4) – 3·(x³/3) + 2·(x²/2) – 5x + c = x⁴ – x³ + x² – 5x + c. Always remember to include ‘+ c’ at the end for an indefinite integral.

求 ∫ (4x³ – 3x² + 2x – 5) dx。逐项积分：4·(x⁴/4) – 3·(x³/3) + 2·(x²/2) – 5x + c = x⁴ – x³ + x² – 5x + c。不定积分最后一定别忘了写 “+ c”。

6. Finding the Constant of Integration | 确定积分常数

If you are given a point on the original curve, such as (x, y) = (2, 10), you can substitute these values into the integrated expression to solve for c. This gives the unique equation of the curve. For example, if dy/dx = 3x² – 2 and the curve passes through (1, 4), then y = x³ – 2x + c; substituting gives 4 = 1 – 2 + c, so c = 5, and y = x³ – 2x + 5.

若已知原曲线上的一点，例如 (x, y) = (2, 10)，可以将其代入积分后的表达式求出 c，从而得到唯一曲线方程。例如，设 dy/dx = 3x² – 2，且曲线通过 (1, 4)，则 y = x³ – 2x + c；代入得 4 = 1 – 2 + c，因此 c = 5，y = x³ – 2x + 5。

7. Definite Integration and Area | 定积分与面积

The definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve y = f(x) and the x-axis from x = a to x = b. Evaluate it by finding an antiderivative F(x), then computing F(b) – F(a). No ‘+ c’ is needed because the constants cancel out.

定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的代数和面积。计算方法是先求一个原函数 F(x)，再求 F(b) – F(a)。无需加 “+ c”，因为常数会抵消。

8. Area Between a Curve and the x-Axis | 曲线与 x 轴之间的面积

To find the actual geometric area (always positive) when the curve crosses the x-axis, split the integral at the roots. Work out the area of each section separately and take absolute values if any part lies below the x-axis. For example, if f(x) is negative on [a, b], then ∫ₐᵇ f(x) dx will be negative, so the true area is -∫ₐᵇ f(x) dx.

当曲线跨越 x 轴时，要计算真正的几何面积（始终为正），需要在交点处拆分积分区域。分别计算每部分的面积，若某部分在 x 轴下方，则对其定积分取绝对值。例如，若 f(x) 在 [a, b] 上为负，则 ∫ₐᵇ f(x) dx 为负值，真实面积是 -∫ₐᵇ f(x) dx。

9. Area Between Two Curves | 两条曲线之间的面积

The area between two curves y = f(x) and y = g(x) from x = a to x = b, where f(x) ≥ g(x) on [a, b], is given by ∫ₐᵇ [f(x) – g(x)] dx. First find the points of intersection to determine the limits a and b, then integrate the difference of the upper and lower functions.

若在 [a, b] 上 f(x) ≥ g(x)，则两曲线之间的面积为 ∫ₐᵇ [f(x) – g(x)] dx。先求出交点以确定积分限 a 和 b，然后对上方函数减去下方函数的差进行积分。

10. Common Mistakes and Tips | 常见错误与技巧

Don’t forget to increase the power by 1 and divide by the new power. Always add ‘+ c’ for indefinite integrals, but never for definite ones. When finding area, sketch the curve to see whether it crosses the x-axis; if it does, split the integral. Double-check the limits of integration, especially when the region is bounded by both curves and the x-axis.

不要忘记指数加 1 并除以新指数。不定积分一定要加 “+ c”，定积分则绝对不要加。求面积时，先画简图判断曲线是否穿过 x 轴；若穿过，必须分段积分。再检查积分上下限，尤其当区域由两曲线与 x 轴共同围成时。

11. Typical Exam Question Types | 典型考题类型

Exam questions often combine differentiation and integration. For instance, you might be given a gradient function dy/dx and a point, then asked to find the equation of the curve (indefinite integration with c). Another common task is to evaluate a definite integral and then use it to find a shaded area. You may also need to find where a curve cuts the x-axis before calculating the total area.

考题常把微分和积分结合起来。例如，给出梯度函数 dy/dx 和一个点，要求曲线方程（带 c 的不定积分）。另一个常见题型是计算定积分，再利用它求阴影面积。也可能需要先求出曲线与 x 轴的交点，再计算总面积。

12. Summary and Final Checklist | 总结与考前清单

Make sure you can confidently integrate xⁿ (n ≠ -1), handle constant multiples and sums, find the constant c from a given point, evaluate definite integrals using F(b) – F(a), and apply integration to area problems with careful handling of regions below the x-axis. Practice with past papers to recognise patterns and avoid careless errors.

确保你能熟练积分 xⁿ（n ≠ -1），处理常数倍与和差，利用已知点求常数 c，用 F(b) – F(a) 计算定积分，并小心处理 x 轴下方的区域，将积分用于面积问题。多做真题，熟悉模式，避免粗心丢分。

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IGCSE Edexcel Maths: Integration Key Points | IGCSE Edexcel 数学：积分 考点精讲