📚 The Definite and Indefinite Integral | 定积分与不定积分
Integration is one of the two central operations in calculus, built to reverse differentiation and to measure accumulation. The indefinite integral recovers a family of functions from a given derivative, while the definite integral computes a precise numerical value representing area, total change, or net accumulation over an interval. Mastering both forms is essential in IB Mathematics, as they underpin everything from kinematics to probability density functions and volumes of revolution.
积分是微积分中的两大核心运算之一，其目的在于逆转微分过程并度量累积量。不定积分可从给定的导数还原出一族函数，而定积分则计算出一个精确的数值，用以表示面积、总量变化或某区间内的净累积。在 IB 数学中，掌握这两种积分形式至关重要，因为它们是运动学、概率密度函数以及旋转体体积等众多知识的基础。
1. Introduction to Integration | 积分引言
At its heart, integration addresses two seemingly different questions: 'Given f '(x), what was f (x)?' and 'What is the area under the curve y = f (x)?' damental Theorem of Calculus, which reveals that differentiation and integration are inverse processes.
积分的核心在于回答两个看似不同的问题：“已知 f '(x)，原来的 f (x) 是什么？”以及“曲线 y = f (x) 下方的面积是多少？”前者导向不定积分（即原函数），后者导向定积分。连接二者的桥梁是微积分基本定理，它揭示了微分与积分互为逆运算。
In IB Mathematics: Analysis and Approaches, integration is treated rigorously, from limit definitions to analytic evaluation. In Applications and Interpretation, the emphasis lies strongly on using integration in modelling contexts, such as finding total revenue from a marginal revenue function or total distance from a velocity function.
在 IB 数学：分析与方法的课程中，从极限定义到解析求值，积分被严格地处理；而在应用与解释课程里，重点则强烈集中于在建模情境中使用积分，例如从边际收入函数求总收入，或从速度函数求总路程。
2. Indefinite Integrals: The Antiderivative | 不定积分：原函数
tion, and the differential dx indicates the variable with respect to which we integrate. The plus C is essential because differentiation destroys constant terms; an entire family of parallel curves shares the same derivative.
函数 f (x) 的不定积分是指任何一个导数等于 f (x) 的函数 F (x)。我们记作 ∫ f (x) dx = F (x) + C，其中 C 为积分常数。符号 ∫ 表示积分，微分 dx 表明积分所针对的变量。加上常数 C 至关重要，因为微分会抹去常数项；整整一族平行的曲线拥有相同的导数。
passes through (0, 5), the specific antiderivative is x³ + 5. In IB questions, a boundary condition or initial condition is often given to determine C.
例如，d/dx (x³) = 3x²，因此 ∫ 3x² dx = x³ + C。若我们还知道曲线经过点 (0, 5)，则特定的原函数为 x³ + 5。在 IB 试题中，通常会给出边界条件或初始条件来确定 C 的值。
3. Basic Rules of Indefinite Integration | 不定积分基本法则
The most fundamental rule is the power rule for integration, which reverses the derivative of xⁿ:
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ -1
最基本的法则是积分的幂法则，它颠倒了 xⁿ 的导数：
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，n ≠ -1
because the denominator becomes zero. In that special case, we use the natural logarithm:
∫ x⁻¹ dx = ∫ (1/x) dx = ln |x| + C
当 n = -1 时，幂法则失效，因为分母会变成零。在这种特殊情况下，我们使用自然对数：
∫ x⁻¹ dx = ∫ (1/x) dx = ln |x| + C
Additional foundational rules include: ∫ eˣ dx = eˣ and sum rules, ∫ k·f (x) dx = k ∫ f (x) dx, ∫ [f (x) ± g(x)] dx = ∫ f (x) dx ± ∫ g(x) dx.
其他基础法则包括：∫ eˣ dx = eˣ + C；∫ sin x dx = -cos x + C；∫ cos x dx = sin x + C；以及常数倍法则与和差法则：∫ k·f (x) dx = k ∫ f (x) dx，∫ [f (x) ± g(x)] dx = ∫ f (x) dx ± ∫ g(x) dx。
4. Integration by Substitution (Reverse Chain Rule) | 换元积分法（反向链式法则）
integration technique when facing a composition of functions. If the integrand involves g'(x) f (g(x)), the reverse chain rule applies immediately: ∫ g' where F is an antiderivative of f. In practice, IB students can set u = g(x), then du = g'(x) dx, and rewrite the integral entirely in terms of u.
当面对复合函数时，换元法是最常用的积分技巧。若被积函数包含 g'(x) f (g(x))，反向链式法则可以直接使用：∫ g'(x) f (g(x)) dx = F (g(x)) + C，其中 F 是 f 的原函数。在实际操作中，IB 学生可令 u = g(x)，从而 du = g'(x) dx，并将整个积分完全用 u 表达出来。
remember to convert back to the original variable when presenting an indefinite integral.
例如，计算 ∫ 2x·sin(x² + 3) dx。令 u = x² + 3，则 du = 2x dx，积分变形为 ∫ sin u du = -cos u + C = -cos(x² + 3) + C。请务必记住，在给出不定积分答案时，要换回原变量。
For definite integrals, the substitution technique also works cleanly: change the limits of integration to match the new variable u, so there is no need to return to x. We will revisit this later.
对于定积分，换元法同样可以干净利落地运用：将积分上下限更换为与新变量 u 对应的值，这样便无需再回到 x。我们稍后会重温这一点。
5. More Integration Techniques (Brief) | 更多积分技巧（简介）
ived from the product rule: ∫ u dv = uv - ∫ v du. A typical application is integrating x·eˣ d forms involving inverse trigonometric functions, such as ∫ 1/(1 + x²) dx = arctan x + C.
除换元法外，高水平阶段的 IB 学生还会经常接触到分部积分法，它源自乘积法则：∫ u dv = uv - ∫ v du。典型应用包括积分 x·eˣ 或 ln x dx，此时需要有策略地选取 u（可使用 LIATE 原则：对数函数、反三角函数、代数函数、三角函数、指数函数）。此外，课程还涵盖涉及反三角函数的标准形式，例如 ∫ 1/(1 + x²) dx = arctan x + C。
Partial fraction decomposition is another tool, used to integrate rational functions by splitting them into simpler fractions. In the IB, this is often applied when the denominator factorises into linear or irreducible quadratic terms.
部分分式分解是另一种工具，用于通过将有理函数拆分为更简单的分式来进行积分。在 IB 课程中，当分母可因式分解为线性因式或不可约二次因式时，常常会用到这一方法。
6. Definite Integrals: The Limit of Riemann Sums | 定积分：黎曼和的极限
he sum Σᵢ f (xᵢ*) Δx. The definite integral is the limit of this sum as n → ∞ (or Δx → 0):
∫ₐᵇ f (x) dx = lim n→∞ Σᵢ₌₁ⁿ f (xᵢ*) Δx
定积分被构造为黎曼和的极限，黎曼和通过矩形来近似曲线下的面积。我们将区间 [a, b] 分割成 n 个宽度为 Δx = (b - a)/n 的小区间，在每个小区间内选取样本点 xᵢ*，并构成和式 Σᵢ f (xᵢ*) Δx。定积分便是当 n → ∞（或 Δx → 0）时该和式的极限：
∫ₐᵇ f (x) dx = lim n→∞ Σᵢ₌₁ⁿ f (xᵢ*) Δx
ve and the x-axis from a to b. If f (x) becomes negative, the integral records signed area — tion models accumulation: a small change in a quantity (rate × time) sums to total change.
如果 f (x) ≥ 0，这个极限便给出了从 a 到 b 曲线与 x 轴之间区域的精确面积。如果 f (x) 变成负值，积分记录的就是带符号的面积——轴下方的区域贡献负值。这一定义也说明了为何积分能够对累积进行建模：微小的变化量（速率 × 时间）累加起来便成为总变化量。
7. The Fundamental Theorem of Calculus | 微积分基本定理
n two parts. Part 1 states that if f is continuous on [a, b], then the function F (x) = ∫ₐˣ f (t) dt is differentiable and F '(x) = f (x). This means integration creates an antiderivative.
微积分基本定理（FTC）分两部分将微分与积分紧密联结。第一部分表明，若 f 在 [a, b] 上连续，则函数 F (x) = ∫ₐˣ f (t) dt 可微，且 F '(x) = f (x)。这意味着积分能够生成一个原函数。
Part 2 provides the practical evaluation rule: if F is any antiderivative of f on [a, b], then
∫ₐᵇ f (x) dx = F (b) - F (a)
第二部分提供了实际求值的法则：若 F 是 f 在 [a, b] 上的任意一个原函数，则
∫ₐᵇ f (x) dx = F (b) - F (a)
We commonly write this as [F (x)]ₐᵇ. In IB exams, the FTC is the workhorse for evaluating exact areas, displacements, and accumulated changes analytically, bypassing the need for limits of Riemann sums.
我们通常将此记作 [F (x)]ₐᵇ。在 IB 考试中，基本定理是解析求解精确面积、位移以及累积变化的主力工具，从而无需再求黎曼和的极限。
8. Evaluating Definite Integrals | 计算定积分
mit a, and subtract. For example, ∫₀² (3x² - 2x) dx = [x³ - x²]₀² = (8 - 4) - (0 - 0) = 4.
要使用基本定理计算定积分：首先求出一个原函数 F (x)（忽略 +C），然后代入上限 b 和下限 a，并相减。例如，∫₀² (3x² - 2x) dx = [x³ - x²]₀² = (8 - 4) - (0 - 0) = 4。
If u = g(x), then when x = a, u = g(a); when x = b, u = g(b). The integral becomes ∫_g(a)g(b) f (u) du. This avoids back-substitution and is generally safer. A common mistake is leaving the original x-limits while the integral is written in u.
在定积分中使用换元法时，务必记得更换上下限。若 u = g(x)，则当 x = a 时，u = g(a)；当 x = b 时，u = g(b)。积分变为 ∫_g(a)g(b) f (u) du。这样就无需再回代，通常也更安全。常见的错误是，积分已经用 u 表示，却仍保留着原来 x 的上下限。
9. Properties of Definite Integrals | 定积分的性质
Several algebraic properties make definite integrals easier to handle in theory and application:
Linearity: ∫ₐᵇ [c₁f (x) + c₂g(x)] dx = c₁∫ₐᵇ f (x) dx + c₂∫ₐᵇ g(x) dx
线性性质：∫ₐᵇ [c₁f (x) + c₂g(x)] dx = c₁∫ₐᵇ f (x) dx + c₂∫ₐᵇ g(x) dx
Reversal of limits: ∫ₐᵇ f (x) dx = -∫_ba f (x) dx
上下限互换：∫ₐᵇ f (x) dx = -∫_ba f (x) dx
Additivity over intervals: ∫ₐᵇ f (x) dx = ∫ₐᶜ f (x) dx + ∫_cᵇ f (x) dx for any c.
区间可加性：对于任意 c，∫ₐᵇ f (x) dx = ∫ₐᶜ f (x) dx + ∫_cᵇ f (x) dx
Comparison: If f (x) ≥ g(x) on [a, b], then ∫ₐᵇ f (x) dx ≥ ∫ₐᵇ g(x) dx.
比较性质：若在 [a, b] 上 f (x) ≥ g(x)，则 ∫ₐᵇ f (x) dx ≥ ∫ₐᵇ g(x) dx
These properties are used to split integrals across known regions, handle piecewise functions, and prove inequalities without full computation. The additivity property is particularly useful when the integrand changes definition or when the integral must be split at an asymptote.
这些性质被用来将积分拆分成已知的区域、处理分段函数，并在无需完整计算的情况下证明不等式。可加性在被积函数定义发生改变或积分必须在渐近线处拆分时特别有用。
10. Area Between Curves | 曲线之间的面积
closed between two curves. If f (x) ≥ g(x) for every x in [a, b], the area between them is
A = ∫ₐᵇ [f (x) - g(x)] dx
定积分最实际的应用之一就是求两条曲线之间所围成的面积。若在 [a, b] 上恒有 f (x) ≥ g(x)，则它们之间的面积为
A = ∫ₐᵇ [f (x) - g(x)] dx
When curves intersect, you must find the intersection points to determine the integration limits and check which function is on top over each subinterval. If the roles of top and bottom swap, you must split the integral at the intersection point and use absolute values or swap the subtraction order. In IB questions, sketching the region is always strongly advised.
当曲线相交时，必须先求出交点以确定积分界限，并检查在每个子区间内哪条曲线位于上方。如果上下位置互换，则必须在交点处将积分拆分，并使用绝对值或交换相减顺序。在 IB 试题中，强烈建议始终先画出区域的草图。
11. Definite Integrals and Accumulated Change | 定积分与累积变化
ge in position) from time a to b is ∫ₐᵇ v(t) dt, while the total distance travelled is ∫ₐᵇ |v(t)| t change and total change appears in contexts ranging from population growth to economics.
只要已知变化率，定积分就能对累积变化进行建模。若质点沿直线以速度 v(t) 运动，从时刻 a 到 b 的位移（位置的净变化）为 ∫ₐᵇ v(t) dt，而走过的总路程则为 ∫ₐᵇ |v(t)| dt。这种净变化与总变化的区分出现在从人口增长到经济学的各种情境中。
In economics, the definite integral of a marginal cost function from 0 to Q gives the total variable cost of producing Q units. Similarly, integrating a rate of water flow into a tank yields the total volume of water added. The definite integral's capacity to multiply varying rates by infinitesimal time increments makes it a universal accumulation engine.
在经济学中，边际成本函数从 0 到 Q 的定积分给出了生产 Q 单位产品的总变动成本。类似地，对流入水箱的水流速率进行积分就可以得到添加的总水量。定积分能够将变化的速率与无限小的时间增量相乘，这使其成为一种通用的累积“引擎”。
12. Avoiding Common Mistakes | 常见错误避免
One of the most persistent errors is forgetting the constant of integration '+C' in indefinite integrals. While it may seem minor, IB mark schemes typically deduct points if C is omitted in a final answer when no boundary condition is given. Always write '+C' unless an initial condition allows you to find its value.
最常见且顽固的错误之一，就是在不定积分中忘记加上积分常数“+C”。这看似小事，但 IB 的评分方案通常规定，若没有给出边界条件而最终答案中缺少 C，就会被扣分。除非初始条件让你能够求出 C 的值，否则请务必写上“+C”。
Another frequent pitfall involves definite integrals with sign errors. When the function lies below the x-axis, the integral gives a negative value. If the question asks for area, you must take the absolute value or integrate the absolute function. Treating a definite integral automatically as 'area' without checking sign leads to wholly incorrect geometric conclusions.
另一个常见的陷阱涉及带符号错误的定积分。当函数位于 x 轴下方时，积分会给出负值。如果题目要求的是面积，就必须取绝对值，或对绝对值函数进行积分。不经符号检查就将定积分自动视为“面积”，会导致完全错误的几何结论。
∫ sin x dx = -cos x + C, or mishandle the limits when substituting. A disciplined approach — ing du, changing limits clearly, and double-checking the antiderivative by differentiating – avoids most mistakes.
学生也常常因遗忘负号而算错三角函数的积分，例如 ∫ sin x dx = -cos x + C，或者在进行换元时错误地处理上下限。训练有素的解题方法——写出 du、清楚地更换上下限，并通过求导来核对原函数——可以避免绝大多数错误。
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