📚 Integration and Differential Equations | 积分与微分方程
Integration is one of the two central ideas of calculus, together with differentiation. It allows us to find areas under curves, compute volumes, solve differential equations, and reconstruct functions from their rates of change. In this article we revise key integration techniques – from basic antiderivatives through substitution and integration by parts – and then apply these skills to solve first-order differential equations, including separable and linear types. We also explore geometric applications such as area between curves and volumes of revolution, building a strong foundation for IB examination success.
积分是微积分两大核心思想之一（另一个是微分），它使我们能够求曲线下的面积、计算体积、求解微分方程并从变化率中还原出原函数。本文我们将复习关键积分技巧——从基本不定积分到换元法和分部积分法——然后把这些技巧应用于求解一阶微分方程，包括可分离变量型和一阶线性方程。我们还会探讨几何应用，如曲线间的面积和旋转体体积，为IB考试打下坚实基础。
1. Antiderivatives and Indefinite Integrals | 原函数与不定积分
tant of integration. The constant is essential because differentiation of a constant gives zero.
函数 f(x) 的不定积分（或原函数）是指满足 F′(x) = f(x) 的函数 F(x)。我们写成 ∫ f(x) dx = F(x) + C，其中 C 为积分常数。常数至关重要，因为常数的导数为零。
ple, ∫ 3x² dx = x³ + C because d/dx (x³) = 3x². Similarly, ∫ cos x dx = sin x + C, and ∫ eˣ dx = eˣ + C. The power rule for integration states that for n ≠ −1, ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C.
例如，∫ 3x² dx = x³ + C，因为 d/dx (x³) = 3x²。类似地，∫ cos x dx = sin x + C，以及 ∫ eˣ dx = eˣ + C。积分的幂法则指出，当 n ≠ −1 时，∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C。
2. Basic Integration Rules and Linearity | 基本积分法则与线性性质
ws us to integrate sums term by term. Combined with standard formulas such as ∫ xⁿ dx, ∫ eˣ dx, ∫ 1/x dx = ln |x| cos x + C, and ∫ cos x dx = sin x + C, we can handle a wide range of elementary functions.
积分是线性运算：∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx。这使得我们可以逐项积分。结合标准公式如 ∫ xⁿ dx、∫ eˣ dx、∫ 1/x dx = ln |x| + C、∫ sin x dx = −cos x + C、∫ cos x dx = sin x + C，我们能够处理大量基本函数。
(4x² + 12x + 9) dx and then integrated term by term. Integrals involving radicals such as √x can be written as x½ and integrated using the power rule.
同样重要的是，积分前往往需要重写被积函数。例如，∫ (2x+3)² dx 可展开为 ∫ (4x² + 12x + 9) dx，然后逐项积分。含有根号如 √x 的积分可写成 x½，再用幂法则积分。
3. Definite Integrals and the Area Under a Curve | 定积分与曲线下的面积
nd integration: if F is any antiderivative of f on [a, b], then ∫ₐᵇ f(x) dx = F(b) − F(a).
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 的净有向面积。微积分基本定理将微分与积分联系起来：若 F 是 f 在 [a, b] 上的任一原函数，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。
To find the area bounded by a curve and the x-axis, we must consider intervals where f(x) is negative: the integral counts that area as negative. The true geometric area is found by integrating the absolute value |f(x)|, or by splitting the interval at zeros of f and taking absolute areas separately.
要求由曲线与 x 轴围成的面积，必须考虑 f(x) 为负的区间：积分会把该部分面积计为负值。真正的几何面积需对 |f(x)| 积分，或在 f 的零点处分段求绝对值面积。
4. Integration by Substitution (Reverse Chain Rule) | 换元积分法（链式法则逆向）
g′(x) dx becomes ∫ f(u) du. After integrating with respect to u, we substitute back to x.
换元积分法是逆向运用链式法则的基本技巧。设 u = g(x)，则 du = g′(x) dx，积分 ∫ f(g(x)) g′(x) dx 变为 ∫ f(u) du。对 u 积分后，再代回 x。
r definite integrals, the limits of integration must also be changed to u-values, eliminating the need to return to x before evaluating.
一个经典例子是 ∫ 2x cos(x²) dx。令 u = x²，du = 2x dx，则原式变为 ∫ cos u du = sin u + C = sin(x²) + C。对于定积分，必须同时将积分限换为 u 值，这样计算时就不需先换回 x。
5. Integration by Parts | 分部积分法
n follow the LIATE guideline (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to select u as the function whose derivative simplifies.
分部积分法是乘积求导法则的积分对应。公式为 ∫ u dv = uv − ∫ v du。选择 u 和 dv 时，常遵循 LIATE 准则（对数、反三角、代数、三角、指数函数），优先选择求导后能简化的函数作为 u。
For example, ∫ x eˣ dx: let u = x, dv = eˣ dx; then du = dx, v = eˣ. Hence ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ iple applications of integration by parts are needed, or cyclic integrals appear as in ∫ eˣ sin x dx.
例如，∫ x eˣ dx：令 u = x，dv = eˣ dx；则 du = dx，v = eˣ。因此 ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。有时需多次使用分部积分，或出现循环积分，如 ∫ eˣ sin x dx。
6. Trigonometric Integrals and Identities | 三角函数的积分与恒等式
/2, and sin 2x = 2 sin x cos x. These transform squares and products into integrable sums.
许多积分涉及正弦和余弦的幂次，关键是利用三角恒等式将被积函数改写成更易于积分的形式。常用恒等式包括 sin²x = (1−cos 2x)/2，cos²x = (1+cos 2x)/2，以及 sin 2x = 2 sin x cos x。它们可将平方和乘积转化为可积的和式。
t u = cos x, du = −sin x dx, to obtain ∫ −(1−u²) du. Integrals of the form ∫ tan x dx = ln |sec x| + C and ∫ sec x dx = ln |sec x + tan x| + C should also be memorised.
对于奇次幂，可保留一个因子进行换元。例如，∫ sin³x dx = ∫ (1−cos²x) sin x dx；令 u = cos x，du = −sin x dx，得到 ∫ −(1−u²) du。形如 ∫ tan x dx = ln |sec x| + C 和 ∫ sec x dx = ln |sec x + tan x| + C 也应熟记。
7. First-Order Separable Differential Equations | 一阶可分离变量微分方程
or explicit solution for y in terms of x plus an arbitrary constant. Initial conditions determine the particular solution.
一阶微分方程 dy/dx = f(x, y) 若能写成 g(y) dy = h(x) dx，就是可分离变量的。求解时两边同时积分：∫ g(y) dy = ∫ h(x) dx，得到用 x 表示的关于 y 的隐式或显式解，外加一个任意常数。初始条件决定特解。
ple, dy/dx = xy can be rearranged as (1/y) dy = x dx, provided y ≠ 0. Integrating gives ln |y| = ½ x² + C, so y = ± e½ x²+C = A e½ x², where A = ± eC the initial condition y(0) = 2 is given, then A = 2 and the particular solution is y = 2 e½ x².
例如，dy/dx = xy 可整理为 (1/y) dy = x dx（假设 y ≠ 0）。积分得 ln |y| = ½ x² + C，因此 y = ± e½ x²+C = A e½ x²，其中 A = ± eC。若给定初始条件 y(0) = 2，则 A = 2，特解为 y = 2 e½ x²。
8. First-Order Linear Differential Equations | 一阶线性微分方程
ntire equation by μ(x), and observe that the left side becomes d/dx (μ(x) y). Then integrate both sides to find y.
一阶线性微分方程的形式为 dy/dx + P(x) y = Q(x)。积分因子法提供了系统解法：计算 μ(x) = e^(∫ P(x) dx)，将整条方程乘以 μ(x)，可发现左边变为 d/dx (μ(x) y)。然后两边积分求得 y。
Example: dy/dx + (1/x) y = x² for x > 0. Here P(x) = 1/x, so μ(x) = e∫ 1/x dx = eln x x + y = x³, which is d/dx (x y) = x³. Integrating gives x y = x⁴/4 + C, so y = x³/4 + C/x.
例子：dy/dx + (1/x) y = x²，x > 0。这里 P(x) = 1/x，因此 μ(x) = e∫ 1/x dx = eln x = x。相乘得：x dy/dx + y = x³，即 d/dx (x y) = x³。积分得 x y = x⁴/4 + C，所以 y = x³/4 + C/x。
9. Applications: Area Between Curves | 应用：曲线间的面积
The area bounded by two curves y = f(x) and y = g(x) from x = a to x = b is given by ∫ₐᵇ |f(x) − g(x)| greater, and integrate the difference without absolute value, summing areas of subregions.
由两条曲线 y = f(x) 与 y = g(x) 在 x = a 到 x = b 之间围成的面积由 ∫ₐᵇ |f(x) − g(x)| dx 给出。实际操作时，在上下曲线发生变化的点分段，确定哪个函数更大，然后分别积分差值（去掉绝对值），将各子区间面积相加。
functions of y, say x = u(y) and x = v(y), the area between them from y = c to y = d is ∫ |u(y) − v(y)| Always sketch the region to identify intersections and the correct limits of integration.
对于表示为 y 的函数的情形，如 x = u(y) 与 x = v(y)，在 y = c 到 y = d 之间的面积为 ∫ |u(y) − v(y)| dy。务必先画出草图，确定交点及正确的积分限。
10. Volumes of Revolution | 旋转体体积
otating about the y-axis, V = π ∫ [g(y)]² dy, where the region is expressed in terms of y.
当一个区域绕 x 轴旋转时，所得立体的体积可用圆盘法求出：V = π ∫ₐᵇ [f(x)]² dx，假设区域由 y = f(x)、x 轴、x = a 和 x = b 围成。若绕 y 轴旋转，V = π ∫ [g(y)]² dy，其中区域用 y 表达。
ere f(x) is the outer radius and g(x) is the inner radius. Similarly, for rotation around the y-axis, volumes are computed analogously with functions of y.
若区域落在两条曲线之间并旋转，则使用垫圈法：V = π ∫ₐᵇ ([f(x)]² − [g(x)]²) dx，其中 f(x) 为外半径，g(x) 为内半径。类似地，绕 y 轴旋转时，用 y 的函数按同样方式计算。
11. Initial Value Problems and Modelling | 初值问题与建模
modelled by dP/dt = k P, with P(0) = P₀. Separating gives dP/P = k dt, leading to P = P₀ ekt. Exponential decay (k < 0) describes radioactive decay or cooling.
许多现实问题可用带初始条件的微分方程（初值问题，IVP）建模。例如，人口增长可用 dP/dt = k P 建模，且 P(0) = P₀。分离变量得 dP/P = k dt，从而得到 P = P₀ ekt。指数衰减（k < 0）用于描述放射性衰变或冷却过程。
Other applications include Newton’s law of cooling, mixing problems, and kinematic equations. Solving the differential equation gives a family of curves; the initial condition picks out the specific curve that matches the given data. Always verify that your solution satisfies both the differential equation and the initial condition.
其他应用包括牛顿冷却定律、混合问题以及运动学方程。求解微分方程得到一族曲线；初始条件选出与给定数据匹配的那条特定曲线。务实验证解是否同时满足微分方程和初始条件。
12. Summary and Exam Tips | 总结与考试技巧
Mastering integration and differential equations requires a combination of recognition, technique selection, and careful algebra. Key skills: identify the type of integral (power, exponential, trigonometric, rational), apply the appropriate method (substitution, parts, trigonometric identities, partial fractions), and handle constants and limits precisely. For differential equations, always separate variables where possible, find the integrating factor for linear equations, and never forget the constant of integration.
掌握积分和微分方程需要识别类型、选择合适方法以及细致的代数运算。关键技能：辨认积分类型（幂函数、指数、三角、有理函数），选用合适方法（换元、分部、三角恒等式、部分分式），并精确处理常数和积分限。对于微分方程，应尽可能分离变量，对线性方程求出积分因子，且永远不要遗漏积分常数。
In IB exams, marks are awarded for setting up integrals correctly, showing the change of limits in substitution, and presenting the final answer clearly, often with π or e left as exact values. Practise past paper questions, especially those that combine integration with geometry or modelling. A well‑drawn sketch can guide your choice of limits and methods. Finally, double‑check your antiderivatives by differentiating them mentally.
在IB考试中，正确建立积分表达式、在换元时写出积分限的变化、清楚呈现最终答案（通常保留 π 或 e 的准确值）均有相应分值。多做历年真题，尤其是将积分与几何或建模相结合的题目。画好草图能帮助你确定积分限和方法。最后，通过心算求导来验证你的原函数是否正确。
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