Integration Essentials | 积分考点精讲

📚 Integration Essentials | 积分考点精讲

Mastering integration is central to success in IB and OCR Mathematics. This article breaks down key integration concepts, techniques, and applications, from basic antiderivatives to volumes of revolution and kinematics, providing clear explanations and exam‑focused insights in both English and Chinese.

掌握积分是 IB 和 OCR 数学取得高分的关键。本文从基础的不定积分到旋转体体积和运动学应用，系统梳理核心概念、技巧与常见题型，提供英中双语对照的清晰讲解与应试要点。

1. Indefinite Integrals and the Constant of Integration | 不定积分与积分常数

An indefinite integral represents the family of all antiderivatives of a function. If F'(x) = f(x), then ∫ f(x) dx = F(x) + C, where C is an arbitrary constant. The presence of ‘+ C’ is mandatory; omitting it can cost marks in an exam.

不定积分表示一个函数的所有原函数族。若 F'(x) = f(x)，则 ∫ f(x) dx = F(x) + C，其中 C 为任意常数。写积分结果时必须加上 + C，考试中漏掉会失分。

The constant of integration arises because differentiation eliminates constant terms. For example, ∫ 2x dx = x² + C, and both x² + 5 and x² − 3 have derivative 2x.

积分常数源于微分会消去常数项。例如 ∫ 2x dx = x² + C，而 x² + 5 和 x² − 3 的导数都是 2x。

2. Basic Integration Rules | 基本积分规则

The power rule for integration is the reverse of the derivative power rule: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, for n ≠ −1. When n = −1, the result is ∫ x⁻¹ dx = ln|x| + C.

幂函数的积分规则是导数幂规则的逆运算：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，n ≠ −1。当 n = −1 时，∫ x⁻¹ dx = ln|x| + C。

Constant multiples and sums are integrated termwise: ∫ k·f(x) dx = k·∫ f(x) dx, and ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.

常数倍和和差可以逐项积分：∫ k·f(x) dx = k·∫ f(x) dx，∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。

Common integrals to memorise include ∫ cos x dx = sin x + C, ∫ sin x dx = −cos x + C, ∫ eˣ dx = eˣ + C, and ∫ 1/x dx = ln|x| + C.

常见函数的积分需要熟记，例如 ∫ cos x dx = sin x + C，∫ sin x dx = −cos x + C，∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C。

3. Integration by Substitution | 换元积分法

Substitution is used when an integral contains a function and its derivative. The general form is ∫ f(g(x))·g'(x) dx. Let u = g(x), then du/dx = g'(x), so du = g'(x) dx, transforming the integral into ∫ f(u) du.

当被积函数包含一个函数及其导数时，可用换元积分法。一般形式为 ∫ f(g(x))·g'(x) dx。令 u = g(x)，则 du/dx = g'(x)，因此 du = g'(x) dx，原积分化为 ∫ f(u) du。

For definite integrals, change the limits along with the variable: if x goes from a to b, u goes from g(a) to g(b), so there is no need to substitute back to x.

对于定积分，需要同时置换积分上下限：若 x 从 a 到 b，则 u 从 g(a) 到 g(b)，这样就不必回代 x。

Example: ∫ 2x·(x²+1)³ dx. Set u = x²+1, then du = 2x dx, so the integral becomes ∫ u³ du = (1/4) u⁴ + C = (1/4)(x²+1)⁴ + C.

例题：∫ 2x·(x²+1)³ dx。设 u = x²+1，则 du = 2x dx，积分化为 ∫ u³ du = (1/4) u⁴ + C = (1/4)(x²+1)⁴ + C。

4. Integration by Parts | 分部积分法

Integration by parts is based on the product rule for differentiation. The formula is ∫ u dv = uv − ∫ v du. The key is to choose u and dv wisely so that ∫ v du is simpler than the original integral.

分部积分法基于微分的乘法法则，公式为 ∫ u dv = uv − ∫ v du。关键在于合理选择 u 和 dv，使得 ∫ v du 比原积分更易求解。

For integrals like ∫ x·eˣ dx, let u = x (differentiates to 1) and dv = eˣ dx, so v = eˣ. Then ∫ x·eˣ dx = x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C.

对于 ∫ x·eˣ dx 这类积分，令 u = x（求导为 1），dv = eˣ dx，则 v = eˣ。于是 ∫ x·eˣ dx = x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C。

Sometimes integration by parts must be applied repeatedly, for example with ∫ x²·sin x dx. Choose u = x², then after the first step reduce the power, and apply parts again.

有时需要反复使用分部积分法，例如 ∫ x²·sin x dx。先令 u = x²，第一次分部后降低 x 的幂次，再继续分部。

For some integrals like ∫ eˣ·sin x dx, applying integration by parts twice returns to the original integral, allowing you to solve it algebraically.

对 ∫ eˣ·sin x dx 等积分，两次分部后会重新出现原积分，可通过代数方法求解。

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

The Fundamental Theorem of Calculus connects differentiation and integration. If F'(x) = f(x) and f is continuous on [a, b], then ∫ₐᵇ f(x) dx = F(b) − F(a). This means you evaluate the antiderivative at the upper and lower limits and subtract.

微积分基本定理将微分与积分联系起来。若 F'(x) = f(x) 且 f 在 [a, b] 上连续，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。也就是说，求出原函数后代入上下限并相减。

Definite integrals yield a number representing the signed area under the curve. The constant of integration C cancels out, so it is not needed.

定积分的结果是一个数值，表示曲线下的有号面积。积分常数 C 在计算中会抵消，因此不需要写出。

Properties: ∫ₐᵇ f(x) dx = −∫ᵦᵃ f(x) dx, and ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᵧᵇ f(x) dx for any c between a and b.

定积分性质：∫ₐᵇ f(x) dx = −∫ᵦᵃ f(x) dx，且对于 a 与 b 之间的任意 c，有 ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᵧᵇ f(x) dx。

6. 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 the definite integral A = ∫ₐᵇ f(x) dx, provided f(x) ≥ 0 on [a, b]. If f dips below the x‑axis, the integral gives a negative value, and the geometric area requires taking absolute values or splitting the interval.

曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间围成的面积由定积分 A = ∫ₐᵇ f(x) dx 给出，前提是 f(x) ≥ 0 在 [a, b] 上。若 f 有部分在 x 轴下方，积分值为负，几何面积需取绝对值或分段计算。

For example, the area under y = x² from 0 to 2 is ∫₀² x² dx = [ (1/3)x³ ]₀² = 8/3 square units.

例如，y = x² 在 0 到 2 区间下方的面积为 ∫₀² x² dx = [ (1/3)x³ ]₀² = 8/3 平方单位。

When the curve is given parametrically, IB and OCR may ask for areas using ∫ y dx = ∫ y(t) · x'(t) dt, with limits changed accordingly.

当曲线以参数方程给出时，IB 和 OCR 考试可能要求使用 ∫ y dx = ∫ y(t) · x'(t) dt 求面积，并相应调整积分限。

7. Area Between Curves | 曲线间面积

The area bounded by two curves y = f(x) (upper) and y = g(x) (lower) between x = a and x = b is A = ∫ₐᵇ [f(x) − g(x)] dx. It is essential to determine which function is on top over the interval; sketching the graphs is highly recommended.

两条曲线 y = f(x)（上侧）与 y = g(x)（下侧）在 x = a 到 x = b 之间围成的面积为 A = ∫ₐᵇ [f(x) − g(x)] dx。必须确定在区间内哪条曲线在上方；强烈建议先画出草图。

If the curves intersect, the total area may need to be calculated in pieces, using the intersection points as limits. Always use the absolute difference |f(x) − g(x)| for geometric area.

若曲线相交，总面积可能需要分段计算，以交点为积分限。几何面积时确保使用绝对值 |f(x) − g(x)|。

8. Volume of Revolution | 旋转体体积

When a region is rotated about the x‑axis, the volume is V = π ∫ₐᵇ [f(x)]² dx. For rotation about the y‑axis, use V = π ∫ᵧᵈ [g(y)]² dy, after expressing x in terms of y. This formula comes from summing the areas of infinitesimal discs.

将区域绕 x 轴旋转所得体积为 V = π ∫ₐᵇ [f(x)]² dx。绕 y 轴旋转时，先将 x 表示为 y 的函数，再用 V = π ∫ᵧᵈ [g(y)]² dy。该公式源于无穷小圆盘面积的累加。

For volume generated between two curves, the ‘washer method’ subtracts the inner radius squared: V = π ∫ₐᵇ [R(x)² − r(x)²] dx, where R is the outer radius and r is the inner radius.

对于两曲线之间的区域旋转，采用“垫圈法”，体积公式为 V = π ∫ₐᵇ [R(x)² − r(x)²] dx，其中 R 为外半径，r 为内半径。

IB students also meet volumes of revolution around lines parallel to axes, such as y = h, requiring adjustment of radii to |f(x) − h|.

IB 学生还可能遇到绕平行于坐标轴的直线（如 y = h）旋转的体积，需要将半径调整为 |f(x) − h|。

9. Integrating Trigonometric Functions | 三角函数的积分

Standard integrals include ∫ sin(ax+b) dx = −(1/a)cos(ax+b) + C and ∫ cos(ax+b) dx = (1/a) sin(ax+b) + C. The pattern follows the chain rule in reverse: divide by the coefficient of x inside the argument.

标准三角函数积分包括 ∫ sin(ax+b) dx = −(1/a)cos(ax+b) + C，∫ cos(ax+b) dx = (1/a) sin(ax+b) + C。处理方法是反用链式法则，除以自变量中 x 的系数。

For sec² x, ∫ sec² x dx = tan x + C. Integrals of tan x can be found by writing tan x = sin x / cos x and using substitution u = cos x, yielding ∫ tan x dx = ln|sec x| + C.

∫ sec² x dx = tan x + C。tan x 的积分可通过写成 sin x / cos x 并换元 u = cos x 求得，结果为 ∫ tan x dx = ln|sec x| + C。

Integration of powers of sine and cosine often uses trigonometric identities, such as sin² x = (1 − cos 2x)/2, to reduce the power.

正、余弦高次幂的积分常利用三角恒等式降幂，例如 sin² x = (1 − cos 2x)/2。

10. Integrating Exponential and Logarithmic Functions | 指数与对数函数的积分

The integral of the natural exponential function is simply itself: ∫ eˣ dx = eˣ + C. For ∫ eᵃˣ dx, the result is (1/a) eᵃˣ + C. This inverse chain rule pattern is crucial.

自然指数函数的积分就是其本身：∫ eˣ dx = eˣ + C。对于 ∫ eᵃˣ dx，结果为 (1/a) eᵃˣ + C。这种反链式法则的模式非常重要。

For logarithmic integration, note that ∫ (f'(x)/f(x)) dx = ln|f(x)| + C. This directly yields ∫ tan x dx = −ln|cos x| + C, and is useful for rational functions.

对数积分的关键是 ∫ (f'(x)/f(x)) dx = ln|f(x)| + C。这一形式可直接给出 ∫ tan x dx = −ln|cos x| + C，并广泛应用于有理函数积分。

Integrals of aˣ are handled by rewriting as e^(x ln a): ∫ aˣ dx = aˣ / ln a + C, for a > 0, a ≠ 1.

∫ aˣ dx 可先改写为 e^(x ln a)，得 aˣ / ln a + C，其中 a > 0，a ≠ 1。

11. Applications: Kinematics | 应用：运动学

Integration is fundamental in kinematics for finding displacement from velocity, and velocity from acceleration. If velocity v(t) is given, displacement s(t) = ∫ v(t) dt. If acceleration a(t) is known, v(t) = ∫ a(t) dt. Constants of integration are determined by initial conditions.

积分在运动学中用于由速度求位移，由加速度求速度。若已知速度 v(t)，则位移 s(t) = ∫ v(t) dt。若已知加速度 a(t)，则 v(t) = ∫ a(t) dt。积分常数由初始条件确定。

For motion in a straight line, total distance travelled is the integral of the absolute value of velocity, ∫ |v(t)| dt, over the time interval. This is different from displacement, which is simply ∫ v(t) dt.

在直线运动中，总路程是速度绝对值对时间的积分，即 ∫ |v(t)| dt，这与位移（∫ v(t) dt）不同。

IB exam questions frequently combine integration with initial conditions: e.g., a(t) = 6t, v(0) = 2, s(0) = 0. Integrate twice to obtain s(t) = t³ + 2t.

IB 考题经常将积分与初始条件结合：例如 a(t) = 6t，v(0) = 2，s(0) = 0。积分两次可得 s(t) = t³ + 2t。

12. Tips for Exam Success | 考试成功技巧

Always include the constant of integration ‘+ C’ for indefinite integrals. For definite integrals, show clear substitution of limits and avoid arithmetic slips by double‑checking signs.

不定积分务必写 + C。定积分要清晰展示代入上下限的过程，仔细核对正负号，避免计算错误。

When integrating by substitution, rewrite the entire integral in terms of the new variable, including the ‘dx’ replacement. Do not mix variables in a single line.

使用换元积分时，务必将整个积分用新变量表示，包括 dx 的替换。不要在同一行混用新旧变量。

Draw diagrams for area and volume problems. A quick sketch helps identify which curve is on top, intersection points, and the appropriate radii for revolution.

面对面积与体积问题时，画出示意图。草图能帮助你判断哪条曲线在上方，找到交点，并确定旋转半径。

Finally, practise recognising the pattern f'(x)/f(x) leading to ln, and know when to use parts versus substitution. In IB exam papers, these two techniques are often tested together.

最后，多练习识别 f'(x)/f(x) 形式以直接得出对数积分，并熟练判断何时用分部积分、何时用换元。IB 试卷经常将这两种技巧组合考查。

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Integration Essentials | 积分考点精讲