📚 IB Mathematics: Core Concepts and Basic Methods of Integration | IB数学：积分核心概念与基本方法
Integration is one of the two fundamental operations in calculus, alongside differentiation. In the IB Mathematics curriculum, it appears in both Analysis and Approaches (AA) and Applications and Interpretation (AI), and it is essential for solving problems involving areas, accumulation, and differential equations.
积分是微积分中与微分并列的两大基本运算之一。在IB数学课程中，积分出现在数学分析与方法（AA）以及数学应用与解释（AI）中，是解决面积、累积量以及微分方程等问题的核心工具。
1. What Is an Integral? | 什么是积分？
An integral is a mathematical object that represents the signed area under a curve, or more generally, the accumulation of a quantity. In IB Mathematics, integration is first introduced as the reverse process of differentiation.
积分是一个数学对象，表示曲线下方的有向面积，或者更一般地，表示某个数量的累积。在IB数学中，积分首先被作为微分的逆运算来引入。
If F'(x) = f(x), then we write ∫ f(x) dx = F(x) + C, where C is the constant of integration. This equation defines the indefinite integral of f.
若 F'(x) = f(x)，则我们记作 ∫ f(x) dx = F(x) + C，其中 C 是积分常数。这个等式给出了 f 的不定积分定义。
∫ f(x) dx = F(x) + C ⇔ F'(x) = f(x)
2. Basic Rules of Integration | 积分的基本法则
Integration follows simple linear rules. The power rule states that for any real number n ≠ −1, the integral of xⁿ is xⁿ⁺¹/(n+1), plus an arbitrary constant. A constant factor can also be moved outside the integral.
积分遵循简洁的线性法则。幂法则指出，对任意实数 n ≠ −1，xⁿ 的积分为 xⁿ⁺¹/(n+1)，再加上任意常数。常数因子也可以移到积分号外面。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
In addition, the integral of a sum equals the sum of the integrals, and the integral of a difference equals the difference of the integrals.
此外，和的积分等于积分的和，差的积分等于积分的差。
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∫ k f(x) dx = k ∫ f(x) dx
∫ k f(x) dx = k ∫ f(x) dx
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∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
3. Standard Integrals | 基本积分公式
Memorising the standard integrals is essential for working quickly in IB exams. The following table lists the most common functions and their indefinite integrals.
熟记基本积分公式对于在IB考试中快速解题至关重要。下表列出了最常见的函数及其不定积分。
| f(x) | ∫ f(x) dx |
| eˣ | eˣ + C |
| 1/x | ln|x| + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
| 1/(1+x²) | arctan x + C |
| 1/√(1−x²) | arcsin x + C |
Notice that the integral of xⁿ for n = −1 is a special case because it produces the natural logarithm.
注意 n = −1 时 xⁿ 的积分是一个特例，它产生自然对数。
4. Integration by Substitution | 换元积分法
Substitution is the reverse of the chain rule for differentiation. It is used when the integrand has the form f(g(x))·g'(x). We set u = g(x), so that du = g'(x) dx.
换元积分法是微分中链式法则的逆运算。它适用于被积函数具有 f(g(x))·g'(x) 的形式时。我们令 u = g(x)，从而 du = g'(x) dx。
∫ f(g(x))·g'(x) dx = ∫ f(u) du = F(u) + C
For example, to evaluate ∫ 2x·e^(x²) dx, let u = x², so du = 2x dx. The integral becomes ∫ eᵘ du = eᵘ + C = e^(x²) + C.
例如，计算 ∫ 2x·e^(x²) dx 时，令 u = x²，则 du = 2x dx。原积分变为 ∫ eᵘ du = eᵘ + C = e^(x²) + C。
5. Integration by Parts | 分部积分法
Integration by parts is the reverse of the product rule. Its formula is:
分部积分法是乘积法则的逆运算，其公式为：
∫ u dv = u v − ∫ v du
We choose u so that its derivative is simpler than u, and dv so that it can be easily integrated. A common mnemonic is LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.
我们选择 u 时，希望它的导数比 u 更简单；选择 dv 时，希望它容易被积分。常用记忆法为 LIATE：对数函数、反三角函数、代数函数、三角函数、指数函数。
For example, to compute ∫ x·eˣ dx, choose u = x and dv = eˣ dx. Then du = dx and v = eˣ, so the result is x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C.
例如，计算 ∫ x·eˣ dx，取 u = x，dv = eˣ dx，则 du = dx，v = eˣ，所以结果为 x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C。
6. Definite Integrals and the Fundamental Theorem of Calculus | 定积分与微积分基本定理
A definite integral has lower and upper limits and evaluates to a number. The Fundamental Theorem of Calculus links differentiation and integration by stating that if F is an antiderivative of f, then the definite integral from a to b equals F(b) − F(a).
定积分带有下限和上限，其结果为数值。微积分基本定理将微分与积分联系起来：若 F 是 f 的原函数，则从 a 到 b 的定积分等于 F(b) − F(a)。
∫ab f(x) dx = F(b) − F(a)
When calculating definite integrals, the constant of integration cancels out, so we do not need to include it.
计算定积分时，积分常数会相互抵消，因此无需写出。
7. Area Under a Curve | 曲线下方的面积
For a continuous function f(x) ≥ 0 on the interval [a, b], the area under the curve from a to b is given by the definite integral:
对于在区间 [a, b] 上连续且满足 f(x) ≥ 0 的函数，曲线从 a 到 b 下方的面积为：
Area = ∫ab f(x) dx
If f(x) is negative on part of the interval, the integral gives a signed area. To find the total geometric area, we split the interval at the x-intercepts and integrate the absolute value of the function.
如果 f(x) 在区间的一部分上为负，积分给出的是有向面积。为了求总几何面积，我们需要在 x 轴交点处分段，并对函数的绝对值进行积分。
8. Area Between Two Curves | 两条曲线之间的面积
When calculating the area enclosed by two curves y = f(x) and y = g(x), we subtract the lower curve from the upper curve. If f(x) ≥ g(x) for all x in [a, b], then the area is:
当计算由两条曲线 y = f(x) 和 y = g(x) 围成的面积时，我们用上方曲线减下方曲线。若在 [a, b] 上 f(x) ≥ g(x)，则面积为：
Area = ∫ab [f(x) − g(x)] dx
If the curves cross, we must find the intersection points and split the integral accordingly.
如果两条曲线相交，我们应先求出交点，然后分段进行积分。
9. Symmetry Properties of Definite Integrals | 定积分的对称性
Symmetry can significantly simplify definite integrals. If f is an even function, then f(−x) = f(x), and integrating over a symmetric interval [−a, a] gives twice the integral from 0 to a.
对称性可以大大简化定积分的计算。若 f 是偶函数，即 f(−x) = f(x)，那么在对称区间 [−a, a] 上的积分等于从 0 到 a 的积分的两倍。
∫−aa f(x) dx = 2 ∫0a f(x) dx
If f is an odd function, then f(−x) = −f(x), and the integral over [−a, a] is zero.
若 f 是奇函数，即 f(−x) = −f(x)，则在 [−a, a] 上的积分为零。
∫−aa f(x) dx = 0
10. Common Mistakes and Study Tips | 常见错误与学习建议
The following list highlights frequent errors made by IB students and some tips to avoid them.
以下列出了IB学生常犯的错误，以及避免这些错误的一些建议。
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Forgetting to add the constant of integration C when evaluating indefinite integrals.
计算不定积分时忘记加上积分常数 C。
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When using substitution in a definite integral, forgetting to change the limits of integration.
在定积分中使用换元法时，忘记改写积分上下限。
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