📚 IB Core Integration | IB 核心积分
Integration is a cornerstone of the IB Mathematics Core syllabus, providing essential tools for computing areas, volumes, and solving motion problems. Mastering the antiderivative concept, learning techniques such as substitution and integration by parts, and linking definite integrals to physical quantities are critical for success in both Analysis & Approaches and Applications & Interpretation examinations.
积分是 IB 数学核心大纲的基石，它为计算面积、体积以及解决运动问题提供了必不可少的工具。掌握原函数概念、学习换元法和分部积分等技巧、并将定积分与物理量联系起来，是在分析与方法和应用与解释两类考试中取得成功的关键。
1. Indefinite Integrals and Basic Rules | 不定积分与基本法则
tiderivatives of a function. Every antiderivative differs by a constant, so the general form always includes the constant of integration + C.
不定积分记作 ∫ f(x) dx，表示一个函数的原函数族。所有原函数之间仅相差一个常数，因此通解必须包含积分常数 + C。
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1
or most polynomial integrations. When n = −1, the integral leads to the natural logarithm.
幂函数法则是大多数多项式积分的基础。当 n = −1 时，积分结果引入自然对数。
Other elementary forms include ∫ k dx = kx + C, ∫ eˣ dx = eˣ must be committed to memory as they appear frequently in substitution and parts problems.
其他基本积分形式包括 ∫ k dx = kx + C、∫ eˣ dx = eˣ + C、∫ sin x dx = −cos x + C 和 ∫ cos x dx = sin x + C。这些公式必须牢记，它们在换元法和分部积分中频繁出现。
2. Integration by Substitution | 代换积分法
tution reverses the chain rule of differentiation. For an integral of the form ∫ f(g(x))·g'(x) dx, we set u = g(x) so that du = g'(x) dx, transforming the integral into ∫ f(u) du which is often easier to evaluate.
换元法逆用微分链式法则。对于形如 ∫ f(g(x))·g'(x) dx 的积分，令 u = g(x)，那么 du = g'(x) dx，积分就化为 ∫ f(u) du，通常更容易求解。
Example: ∫ (3x² + 2x)(x³ + x²)⁴ dx
then du = (3x² + 2x) dx. The integral becomes ∫ u⁴ du = (1/5)u⁵ + C = (1/5)(x³ + x²)⁵ + C.
令 u = x³ + x²，则 du = (3x² + 2x) dx。积分变为 ∫ u⁴ du = (1/5)u⁵ + C = (1/5)(x³ + x²)⁵ + C。
In definite integrals, remember to change the limits of integration to u-values. This avoids the need to back-substitute and simplifies the arithmetic.
在定积分中，必须将积分上下限替换为 u 值。这样可避免回代原变量，简化计算。
3. Integration by Parts | 分部积分法
presented as ∫ u dv = uv − ∫ v du, where u and dv are chosen from the original integrand.
分部积分法源自微分的乘法法则，公式为 ∫ u dv = uv − ∫ v du，其中 u 和 dv 从原始被积函数中选取。
∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx
A common mnemonic for choosing u is LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Differentiate u and integrate dv to produce a simpler integral on the right-hand side.
选择 u 的通用助记法是 LIATE：对数函数、反三角函数、代数函数、三角函数、指数函数。对 u 求导，对 dv 积分，使右侧的积分更简单。
Example: ∫ x eˣ dx. Choose u = x (algebraic), dv = eˣ dx. Then du = dx, v = eˣ, giving x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.
示例：∫ x eˣ dx。选取 u = x（代数函数），dv = eˣ dx。则 du = dx，v = eˣ，得到 x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。
es parts must be applied twice or combined with algebraic manipulation, especially for ∫ eˣ sin x dx type integrals.
有时需要两次分部积分或结合代数变换，尤其是对 ∫ eˣ sin x dx 这类积分。
4. Definite Integrals and the Fundamental Theorem | 定积分与微积分基本定理
The definite integral ∫ₐᵇ f(x) dx computes the net signed area between the curve y = f(x) and the x‑ to x = b. The Fundamental Theorem of Calculus links antiderivatives to definite integrals.
定积分 ∫ₐᵇ f(x) dx 计算曲线 y = f(x) 与 x 轴在区间 [a, b] 上的代数和面积。微积分基本定理将原函数与定积分联系起来。
∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x)
To evaluate a definite integral, first find an antiderivative F(x), then substitute the upper and lower bounds and subtract. The constant + C cancels out, which is why it is omitted in definite calculations.
计算定积分时，首先求出原函数 F(x)，再代入上下限相减。常数 + C 在相减过程中消失，这就是定积分计算中不写常数的原因。
) dx ± ∫ₐᵇ g(x) dx, and ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx are essential for simplifying problems.
定积分的性质，如 ∫ₐᵇ k·f(x) dx = k·∫ₐᵇ f(x) dx、∫ₐᵇ [f(x)±g(x)] dx = ∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx 以及 ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx，对于化简题目至关重要。
5. Area Under a Curve | 曲线下方面积
The area bounded by y = f(x), the x‑axis, and the vertical lines x = a and x = b is given by Area = ∫ₐᵇ |f(x)| dx if the curve crosses the axis. For regions where f(x) ≥ 0, the area is simply ∫ₐᵇ f(x) dx.
由曲线 y = f(x)、x 轴以及竖直线 x = a 和 x = b 围成的面积，如果曲线穿过 x 轴，则由 Area = ∫ₐᵇ |f(x)| dx 给出。对于 f(x) ≥ 0 的区域，面积就是 ∫ₐᵇ f(x) dx。
When f(x) changes sign within [a, b], split the integral into subintervals where the function is entirely above or entirely below the axis, take the absolute value of each piece, and sum them.
若 f(x) 在 [a, b] 内改变符号，需将积分拆分为函数全程在轴上方和全程在下方的子区间，分别取每段的绝对值再求和。
Area = ∫ₐᶜ f(x) dx − ∫ᶜᵇ f(x) dx, for f(x) ≤ 0 on [c, b]
Many IB questions require identifying where the curve meets the x‑axis by solving f(x) = 0, then constructing a piecewise integral.
许多 IB 题目需要先通过解方程 f(x) = 0 找出曲线与 x 轴的交点，然后分段构建积分。
6. Area Between Curves | 两曲线间面积
= f(x) (upper) and y = g(x) (lower) intersect at x = a and x = b, the area between them is ∫ₐᵇ [f(x) − g(x)] dx. Always subtract the lower curve from the upper curve over the entire interval.
若两曲线 y = f(x)（上方）和 y = g(x)（下方）交于 x = a 和 x = b，则它们之间的面积为 ∫ₐᵇ [f(x) − g(x)] dx。在整个区间内都要用上方曲线减去下方曲线。
To determine which function is uppermost, either sketch the graphs or test a point between intersection points. If the curves cross, the area must be split at the crossover points.
要判断哪条曲线在上方，可以画草图或在交点之间取测试点。若曲线相交叉，需要在交点处拆分面积积分。
y (e.g., x = f(y)), the area formula becomes ∫ₐᵇ [right − left] dy, where the limits are y‑ approach is particularly useful when functions are easier to integrate with respect to y.
如果边界以 y 的函数形式给出（如 x = f(y)），面积公式变为 ∫ₐᵇ [右 − 左] dy，其中积分限为 y 坐标。当函数对 y 积分更容易时，这种方法特别有用。
7. Volume of Revolution | 旋转体体积
When a curve y = f(x) from x = a to x = b is rotated about the x‑axis, the volume of the solid formed is calculated by the disc method.
当曲线 y = f(x) 在 x = a 到 x = b 之间绕 x 轴旋转时，所形成的立体体积用圆盘法计算。
V = π ∫ₐᵇ [f(x)]² dx
Each cross‑section perpendicular to the x‑axis is a circle of radius f(x), giving area π [f(x)]². Integrating this area along the axis yields the total volume.
垂直于 x 轴的每个截面都是一个半径为 f(x) 的圆，面积为 π [f(x)]²。沿轴对该面积积分即得总体积。
For rotation about the y‑ y = d, use V = π ∫₆ᵈ [g(y)]² dy. Always ensure the region is clearly defined without gaps.
若绕 y 轴旋转，边界为 x = g(y)，区间为 y = c 到 y = d，则使用 V = π ∫₆ᵈ [g(y)]² dy。务必确保旋转的区域清晰无缺口。
s is required; use the washer method: V = π ∫ₐᵇ ([f(x)]² − [g(x)]²) dx, where f(x) ≥ g(x).
有些题目要求计算两曲线之间区域旋转生成的体积，此时采用圆环法：V = π ∫ₐᵇ ([f(x)]² − [g(x)]²) dx，其中 f(x) ≥ g(x)。
8. Kinematics Application | 运动学应用
ed by s(t) = ∫ v(t) dt, with the constant of integration determined by initial conditions.
积分将直线运动中的位移、速度与加速度联系起来。给定速度 v(t)，位移 s(t) 由 s(t) = ∫ v(t) dt 得出，积分常数由初始条件确定。
nce travelled, however, is the accumulation of speed regardless of direction: Distance = ∫ |v(t)| over the time interval. This is often evaluated by splitting the integral where v(t) = 0.
而总路程是速率与时间乘积的累积，与方向无关：Distance = ∫ |v(t)| dt。通常需要先求出 v(t) = 0 的时刻，再分段积分。
). This demonstrates the fundamental role of integration in deriving equations of motion from basic principles.
加速度 a(t) 积分可得速度：v(t) = ∫ a(t) dt + v(0)。这展示了积分从基本原理推导运动方程的核心作用。
9. Integrating Trigonometric and Exponential Functions | 三角函数与指数函数的积分
t x + C, and ∫ sec x tan x dx = sec x + C. These arise in substitution and parts problems.
除了基本的正弦与余弦积分，IB 核心要求熟悉 ∫ sec² x dx = tan x + C、∫ csc² x dx = −cot x + C 以及 ∫ sec x tan x dx = sec x + C。这些常出现在换元或分部积分题中。
For exponential functions, ∫ aˣ dx = aˣ/ln a + C, a > 0, a ≠ 1. The natural exponential remains the simplest: ∫ eˣ dx = eˣ + C, and ∫ eᵏˣ dx = (1/k)eᵏˣ + C through substitution.
对于指数函数，∫ aˣ dx = aˣ/ln a + C，其中 a > 0 且 a ≠ 1。自然指数函数依然最简单：∫ eˣ dx = eˣ + C，而 ∫ eᵏˣ dx = (1/k)eᵏˣ + C 可通过换元得出。
Compounds such as ∫ e²ˣ arts, often leading back to the original integral, which can then be solved algebraically.
复合型积分如 ∫ e²ˣ sin x dx 需要反复使用分部积分，通常会回到原积分，此时可借助代数方法求解。
10. Tips and Common Pitfalls | 技巧与常见误区
Always add + C in indefinite integrals. For definite integrals, double-check that the antiderivative is correct by differentiating, and confirm that the limits are substituted in the correct order: F(upper) − F(lower).
不定积分永远要加 + C。对于定积分，要通过求导验证原函数是否正确，并确认代入积分限的顺序为 F(上限) − F(下限)。
dx entirely in terms of du. Write out du = ... dx and then solve for dx to avoid mistakes.
在换元法中，许多学生忘记更换积分限或未将 dx 完全用 du 表达。应写出 du = ... dx 再解出 dx，以避免错误。
When calculating area, sketch the region and identify intersections precisely. Using a calculator to find intersection points is allowed but always interpret the sketch to set up the correct integral.
计算面积时，先画草图并准确找出交点。虽然可使用计算器求交点，但务必根据草图设定正确的积分区间。
For volumes, remember to square the function before integrating, and never square the integral. Misplacing the square is one of the most frequent errors in exam scripts.
计算体积时，记得先平方函数再积分，切勿对积分结果平方。平方位置的错置是试卷中最常见的错误之一。
Finally, practice mixing techniques: a single problem may require substitution followed by parts or combining trigonometric identities. The IB exam often tests integration in unfamiliar contexts, so build flexibility.
最后，要多练习混合技巧：一道题可能先换元再分部，或者结合三角恒等式。IB 考试常常在陌生情境下考查积分，应当培养灵活应变的能力。
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