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

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

Integration is a cornerstone of CIE IGCSE Mathematics, bridging differentiation and practical applications like area and kinematics. This guide unpacks every essential concept, from reverse differentiation to definite integrals, with clear rules, worked examples, and common pitfalls. Mastering integration will boost your confidence and exam performance.

积分是 CIE IGCSE 数学的核心内容，它将微分与实际应用（如面积和运动学）紧密相连。本文逐一剖析从逆微分到定积分的每个重要概念，配以清晰法则、范例和常见错误，助你掌握积分、提升考试表现。

1. Understanding Integration as Reverse Differentiation | 理解积分为微分的逆运算

Integration is the process of finding an antiderivative. If F'(x) = f(x), then ∫ f(x) dx = F(x) + C, where C is an arbitrary constant called the constant of integration. This arises because the derivative of any constant is zero, so we must account for all possible original functions.

积分是寻找反导数的过程。若 F'(x) = f(x)，则 ∫ f(x) dx = F(x) + C，其中 C 是任意常数，称为积分常数。这是因为任何常数的导数都是零，所以必须加上 C 以涵盖所有可能的原函数。

For example, since d/dx (x²) = 2x, we know ∫ 2x dx = x² + C. Always remember the +C in indefinite integrals, as it is a very common mark in exams.

例如，因为 d/dx (x²) = 2x，我们知道 ∫ 2x dx = x² + C。在不定期积分中永远记得加上 +C，这是考试中非常常见的得分点。

2. Basic Integration Rule: The Power Rule | 基本积分法则：幂法则

For any real number n ≠ -1, the power rule for integration states:

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

对于任意实数 n ≠ -1，积分幂法则为：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

Notice that we increase the power by 1 and divide by the new power. This is the exact opposite of the differentiation power rule.

注意我们将指数增加 1，再除以新指数。这正是微分幂法则的逆过程。

Example: ∫ x³ dx = x⁴/4 + C. Example: ∫ √x dx = ∫ x^½ dx = x^3/2/(3/2) + C = (2/3)x^3/2 + C. Always rewrite roots and fractions as powers before integrating.

例如：∫ x³ dx = x⁴/4 + C。例如：∫ √x dx = ∫ x^½ dx = x^3/2/(3/2) + C = (2/3)x^3/2 + C。积分前务必将根式和分式改写为指数形式。

3. Integrating Constants and Sums/Differences | 积分常数与和差函数

The integral of a constant k with respect to x is simply kx + C. Also, the integral of a sum or difference is the sum or difference of the integrals.

常数 k 的积分就是 kx + C。同样，和差函数的积分等于积分的和差。

∫ k dx = kx + C

∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx

These properties allow us to integrate term by term. For instance, ∫ (4x³ – 2x + 5) dx = x⁴ – x² + 5x + C. Every term follows its own power rule, and constants combine into a single +C at the end.

这些性质允许我们逐项积分。例如，∫ (4x³ – 2x + 5) dx = x⁴ – x² + 5x + C。每一项遵循各自的幂法则，最后所有常数合并为一个 +C。

4. Integrating (ax + b)ⁿ | 积分 (ax+b)ⁿ

When the integrand is a linear expression raised to a power, we use the reverse chain rule. Divide by the coefficient of x and also by the new power.

∫ (ax + b)ⁿ dx = (1/a) · (ax + b)ⁿ⁺¹/(n+1) + C, n ≠ -1

当被积函数是线性表达式的乘方时，我们使用反链式法则。要除以 x 的系数，并除以新指数。

∫ (ax + b)ⁿ dx = (1/a) · (ax + b)ⁿ⁺¹/(n+1) + C, n ≠ -1

Example: ∫ (2x + 3)⁴ dx = (1/2) · (2x + 3)⁵/5 + C = (2x + 3)⁵/10 + C. Always remember the factor 1/a; a common mistake is to omit it.

例子：∫ (2x + 3)⁴ dx = (1/2) · (2x + 3)⁵/5 + C = (2x + 3)⁵/10 + C。切莫忘记系数 1/a；漏掉它是常见错误。

5. Integrating Exponential Functions | 积分指数函数

For the exponential function with base e, integration again requires division by the coefficient of x in the exponent.

∫ e^ax+b dx = (1/a) e^ax+b + C

对于以 e 为底的指数函数，积分时同样要除以指数中 x 的系数。

∫ e^ax+b dx = (1/a) e^ax+b + C

Example: ∫ e^5x-2 dx = (1/5)e^5x-2 + C. If the exponent is just x, ∫ e^x dx = e^x + C. These integrals appear frequently in growth/decay problems.

例如：∫ e^5x-2 dx = (1/5)e^5x-2 + C。若指数仅为 x，∫ e^x dx = e^x + C。这些积分经常出现于增长/衰减问题中。

6. Integrating Trigonometric Functions | 积分三角函数

The integrals of basic sine and cosine functions follow from their derivatives, again adjusting for the coefficient of x inside the function.

∫ sin(ax + b) dx = –(1/a) cos(ax + b) + C

∫ cos(ax + b) dx = (1/a) sin(ax + b) + C

基本正弦和余弦函数的积分源自它们的导数，同样要调整函数内部 x 的系数。

∫ sin(ax + b) dx = –(1/a) cos(ax + b) + C

∫ cos(ax + b) dx = (1/a) sin(ax + b) + C

Note the negative sign when integrating sine. Example: ∫ cos(3x – π) dx = (1/3) sin(3x – π) + C. Always check your answer by differentiating; it’s the best way to catch sign errors.

注意积分正弦时出现的负号。例子：∫ cos(3x – π) dx = (1/3) sin(3x – π) + C。务必通过微分检验答案，这是发现符号错误的最佳方法。

7. Integrating 1/(ax + b) | 积分 1/(ax+b)

The power rule fails for n = -1, but this important case leads to the natural logarithm.

∫ 1/(ax + b) dx = (1/a) ln |ax + b| + C

当 n = -1 时幂法则失效，但这个重要情形导向自然对数。

∫ 1/(ax + b) dx = (1/a) ln |ax + b| + C

The absolute value ensures the argument of ln is positive, as the logarithm is only defined for positive numbers. Example: ∫ 1/(2x-1) dx = (1/2) ln |2x-1| + C. If the denominator is just x, ∫ 1/x dx = ln |x| + C.

绝对值保证 ln 的参数为正，因为对数只对正数有定义。例子：∫ 1/(2x-1) dx = (1/2) ln |2x-1| + C。若分母仅为 x，∫ 1/x dx = ln |x| + C。

8. Definite Integrals and Evaluating Limits | 定积分与上下限计算

A definite integral computes the net area between a curve and the x-axis from a lower limit a to an upper limit b. It is evaluated using the fundamental theorem of calculus.

∫_a^b f(x) dx = [F(x)]_a^b = F(b) – F(a)

定积分计算曲线与 x 轴之间从下限 a 到上限 b 的净面积。它利用微积分基本定理求值。

∫_a^b f(x) dx = [F(x)]_a^b = F(b) – F(a)

First find the antiderivative F(x) (without +C), substitute the upper limit, then subtract the value at the lower limit. Example: ∫₁³ 2x dx = [x²]₁³ = 9 – 1 = 8.

先求出反导数 F(x)（不带 +C），代入上限的值，再减去下限的值。例如：∫₁³ 2x dx = [x²]₁³ = 9 – 1 = 8。

Remember that a definite integral can be negative if the curve lies below the x-axis, representing a net signed area. Always take care with the order of subtraction.

记住，如果曲线在 x 轴下方，定积分可能为负值，表示带符号的净面积。减法顺序务必仔细。

9. Finding Area Under a Curve | 求曲线下面积

To find the actual geometric area enclosed by a curve y = f(x) and the x-axis between x = a and x = b, you must consider where the curve crosses the axis.

求曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间所围的实际几何面积时，必须考虑曲线与轴的交点。

If f(x) ≥ 0 on [a,b], then Area = ∫_a^b f(x) dx. If f(x) ≤ 0, the integral gives a negative value, so Area = –∫_a^b f(x) dx. If the graph crosses the x-axis, split the integral at the intercepts and take absolute values of each piece, or sum the positive areas reflected.

若在 [a,b] 上 f(x) ≥ 0，则 Area = ∫_a^b f(x) dx。若 f(x) ≤ 0，积分值为负，故 Area = –∫_a^b f(x) dx。若图像穿过 x 轴，需在交点处拆分积分，并对每一段取绝对值，或将反射为正的面积相加。

Always sketch the graph if possible, and double-check that your final area is a positive number. Exam questions often penalise a negative area given as the final answer.

尽可能画草图，并仔细确认最终面积是正数。考试中若最终答案给出负面积，往往会被扣分。

10. Area Between Two Curves | 两曲线之间的面积

The area enclosed between two curves y = f(x) (upper) and y = g(x) (lower) from x = a to x = b is given by:

Area = ∫_a^b [f(x) – g(x)] dx

两条曲线 y = f(x)（上）和 y = g(x)（下）在 x = a 到 x = b 之间所围的面积公式为：

Area = ∫_a^b [f(x) – g(x)] dx

Identify which curve is uppermost on the interval; sometimes you must find intersection points to set the limits a and b. Simplify f(x) – g(x) before integrating.

确定在区间内哪条曲线在上方；有时须先求出交点以确定上下限 a 和 b。积分前先化简 f(x) – g(x)。

Example: Find the area between y = x² and y = x from x = 0 to x = 1. The line y = x is above y = x² on (0,1), so Area = ∫₀¹ (x – x²) dx = [x²/2 – x³/3]₀¹ = 1/2 – 1/3 = 1/6.

例如：求 y = x² 与 y = x 在 x = 0 到 1 之间的面积。在 (0,1) 内直线 y = x 在 y = x² 上方，故 Area = ∫₀¹ (x – x²) dx = [x²/2 – x³/3]₀¹ = 1/2 – 1/3 = 1/6。

11. Applications to Kinematics | 运动学应用

Integration is vital for motion problems. Given acceleration a(t), velocity v(t) is its integral, and displacement s(t) is the integral of velocity, with initial conditions determining the constants.

v(t) = ∫ a(t) dt, s(t) = ∫ v(t) dt

积分对于运动问题至关重要。已知加速度 a(t)，速度为它的积分，位移为速度的积分，并由初始条件确定常数。

v(t) = ∫ a(t) dt, s(t) = ∫ v(t) dt

Example: A particle moves with a(t) = 2t – 1. If v(0) = 3, then v(t) = ∫ (2t – 1) dt = t² – t + C. Using v(0)=3 gives C=3, so v(t) = t² – t + 3. Displacement is found by another integration, typically with a given initial position.

例：质点以 a(t) = 2t – 1 运动。若 v(0) = 3，则 v(t) = ∫ (2t – 1) dt = t² – t + C。由 v(0)=3 得 C=3，故 v(t) = t² – t + 3。位移可通过再次积分求得，通常伴有给定的初始位置。

In CIE questions, you may also need to find total distance travelled by integrating the absolute value of velocity, so careful analysis of when the particle changes direction is required.

在 CIE 考题中，还可能要求通过对速度的绝对值积分求出总路程，因此需要仔细分析质点何时改变方向。

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

Even strong students slip up on these recurring pitfalls. Review them before the exam.

即便是优秀学生也常在这些反复出现的陷阱中失分。考前务必回顾。

Forgetting +C: Always add the constant of integration for indefinite integrals. It often carries a mark.

忘记 +C：不定积分永远记得加上积分常数，这通常占 1 分。

Incorrect coefficient for reverse chain: When integrating (ax+b)ⁿ, e^ax+b, sin(ax+b), or 1/(ax+b), don’t miss the 1/a factor.

反链式系数错误：积分 (ax+b)ⁿ、e^ax+b、sin(ax+b) 或 1/(ax+b) 时，切勿遗漏 1/a 因子。

Negative area answers: For area questions, sketch the graph and ensure your final answer is positive. Split the integral if necessary.

面积为负：面对面积题，画草图，保证最终答案是正数。必要时拆分积分。

Misplacing limits in definite integrals: Always compute F(upper) – F(lower); reversing the order flips the sign.

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