Integration Key Points | 积分考点精讲

📚 Integration Key Points | 积分考点精讲

Integration is the reverse process of differentiation and one of the two central operations in calculus. In the Edexcel A‑Level Mathematics specification, integration appears in both pure and applied contexts: you will be expected to evaluate indefinite and definite integrals analytically, apply integration techniques to find areas and volumes, solve first‑order differential equations, and use numerical methods such as the trapezium rule. The ability to choose the correct method and handle algebraic manipulation with confidence is essential for success in the exam.

积分是微分的逆运算，也是微积分中的两大核心操作之一。在爱德思 A‑Level 数学大纲中，积分同时出现在纯数学和应用数学部分：需要学会解析地计算不定积分和定积分，运用积分技巧求面积与体积，求解一阶微分方程，以及使用梯形法则等数值方法。能否正确选择方法并熟练进行代数整理，是考试成功的关键。

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1. Basic Integration Rules | 基本积分公式

You must memorise the standard integrals listed in the Edexcel formula booklet. For any real constant n ≠ −1, the power rule is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. The integral of 1/x is ln |x| + C. The exponential function integrates to itself: ∫ e^kx dx = (1/k)e^kx + C for any constant k have standard antiderivatives, such as ∫ sin x dx = −cos x + C and ∫ cos x dx = sin x + C.

必须熟记爱德思公式表列出的标准积分。对于任意实常数 n ≠ −1，幂函数的积分公式为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。1/x 的积分是 ln |x| + C。指数函数的积分是自身形式：∫ e^kx dx = (1/k)e^kx + C，其中 k 为常数。三角函数的原函数也有标准结果，如 ∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C。

Function f(x)	Indefinite Integral ∫ f(x) dx
xn (n ≠ −1)	xn+1/(n+1) + C
1/x	ln |x| + C
ex	ex + C
sin x	−cos x + C
cos x	sin x + C
sec2 x	tan x + C
cosec2 x	−cot x + C
sec x tan x	sec x + C
cosec x cot x	−cosec x + C
1/√1−x2 (|x| < 1)	arcsin x + C
−1/√1−x2 (|x| < 1)	arccos x + C
1/(1+x2)	arctan x + C

m: ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx. Always simplify the integrand first – for example, expand brackets or split fractions into separate terms before integrating.

若被积函数是常数倍或若干项的和，可以逐项积分：∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx。应始终先化简被积函数，例如展开括号或将分式拆分成若干项，然后再积分。

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

An indefinite integral represents a family of functions whose derivative is the integrand. The constant of integration C must always be included. Losing the constant is a common mistake that can cost marks, especially when an initial condition is provided later to find a particular solution.

不定积分表示一族函数，其导数等于被积函数。积分常数 C 必须始终写出。漏掉常数是常见错误，当后续给出初始条件以确定特解时，漏掉常数会直接丢分。

For example, ∫ (3x² + 2) dx = x³ + 2x + C. Without the constant, all subsequent work based on the antiderivative (such as evaluating a definite integral or solving a differential equation) may be incomplete or incorrect. Always write +C immediately after finding an indefinite integral.

例如，∫ (3x² + 2) dx = x³ + 2x + C。若没有常数，后续基于原函数的计算（例如计算定积分或解微分方程）可能不完整甚至错误。在求出不定积分后，必须立即加上 +C。

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

The definite integral ∫_a^b al Theorem of Calculus: find any antiderivative F(x) of f(x) and compute F(b) − F(a). The constant C cancels, so it is not needed when using the theorem directly.

定积分 ∫_a^b f(x) dx 表示曲线 y = f(x) 在 x = a 与 x = b 之间的带符号面积。计算时使用微积分基本定理：求出 f(x) 的任一原函数 F(x)，再计算 F(b) − F(a)。此时积分常数 C 会抵消，因此直接使用定理时无需写出 C。

∫_a^b f(x) dx = F(b) − F(a)

Always show the substitution step clearly, e.g., [F(x)]_a^b. Pay attention to signs when subtracting: it is easy to mistakenly reverse the limits. If the integrand is negative over the interval, the definite integral gives a negative value, representing the area below the x‑axis.

务必清晰地写出代入步骤，例如 [F(x)]_a^b。注意减法时的符号，容易将上下限弄反。若被积函数在区间上为负，定积分将得到负值，代表 x 轴下方的面积。

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4. Integration by Substitution | 换元积分法

Substitution is used when a function contains a composite expression, such as (2x+3)⁵ or e^x² substitution or choose one yourself. The chain rule is reversed: let u = g(x), then du = g'(x) dx, and replace all x‑terms with u‑expressions before integrating.

换元法用于被积函数包含复合表达式时，例如 (2x+3)⁵ 或 e^x²。可以使用题目给出的代换，或自己选择一个。本质上是对链式法则的逆用：设 u = g(x)，则 du = g'(x) dx，将所有含 x 的项替换为关于 u 的表达式后再积分。

integral with substitution, you have two options: change the limits to the corresponding u‑ ts often saves time. Always remember to adjust the limits if necessary: when x = a, u = g(a); when x = b, u = g(b).

对于涉及换元的定积分，有两种处理方法：将积分限换成对应的 u 值后用 u 计算，或者求出关于 x 的原函数后再使用原始上下限。两种方法均可，但换限常常能节省时间。若选择换限，务必记得对应关系：当 x = a 时 u = g(a)；当 x = b 时 u = g(b)。

Example: Evaluate ∫ x (x²+1)⁴ dx. Let u = x²+1, then du = 2x dx, so x dx = ½ du. The integral becomes ∫ ½ u⁴ du = ½ × u⁵/5 + C = (x²+1)⁵/10 + C.

例题：计算 ∫ x (x²+1)⁴ dx。令 u = x²+1，则 du = 2x dx，从而 x dx = ½ du。积分变为 ∫ ½ u⁴ du = ½ × u⁵/5 + C = (x²+1)⁵/10 + C。

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5. Integration by Parts | 分部积分法

Integration by parts is derived from the product rule for differentiation and is used when the integrand is a product of two functions of different types, such as x e^x, x sin x, or ln x. The formula is:

分部积分法源自微分的乘法法则，适用于被积函数为两类不同函数乘积的情况，如 x e^x、x sin x 或 ln x。其公式为：

∫ u dv = uv − ∫ v du

rding to the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) – fferentiating u, and v by integrating dv. The resulting integral ∫ v du should be simpler.

实际应用中可根据 LIATE 法则选择 u：对数函数、反三角函数、代数函数、三角函数、指数函数，优先选择求导后能简化的函数作为 u。确定 u 后，剩余部分（包含 dx）作为 dv。接着微分 u 得到 du，积分 dv 得到 v。最终的积分 ∫ v du 应比原积分更简单。

Example: ∫ x e^x dx. Let u = x, dv = e^x dx. Then du = dx, v = e^x. Apply the formula: = x e^x − ∫ e^x dx = x e^x − e^x ln x dx, treat ln x as u, and dv = 1 dx. Then du = (1/x) dx, v = x, giving x ln x − x + C.

例题：∫ x e^x dx。令 u = x，dv = e^x dx，则 du = dx，v = e^x。代入公式：= x e^x − ∫ e^x dx = x e^x − e^x + C。对于 ∫ ln x dx，视 ln x 为 u，dv = 1 dx，则 du = (1/x) dx，v = x，得 x ln x − x + C。

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6. Integration Using Partial Fractions | 部分分式积分法

Rational functions where the numerator has lower degree than the denominator can often be split into partial fractions, making integration straightforward. Common forms include distinct linear factors, repeated linear factors, and irreducible quadratic factors. After decomposing, each term integrates to a log function or an arctangent.

分子次数低于分母的有理函数通常可以拆分成部分分式，从而轻松积分。常见形式包括不同的一次因子、重复的一次因子以及不可约的二次因子。拆解后，每一项可积分为对数函数或反正切函数。

through by the denominator or by substituting convenient values. Then ∫ A/(x+a) dx = A ln |x+a| + C. For a repeated factor (x+a)², include terms A/(x+a) + B/(x+a)², where the second term integrates to −B/(x+a).

对于不同的一次因子，可将 (px+q)/((x+a)(x+b)) 写成 A/(x+a) + B/(x+b)。通过乘开分母或代入简便数值求出 A 与 B。然后 ∫ A/(x+a) dx = A ln |x+a| + C。对于重复因子 (x+a)²，需设项 A/(x+a) + B/(x+a)²，其中第二项积分为 −B/(x+a)。

Example: ∫ (2x)/(x² ). Set 2x/[(x−1)(x+1)] = A/(x−1) + B/(x+1). Solve to get A=1, B=1. The integral becomes ln |x−1| + ln |x+1| + C = ln |x²−1| + C.

例题：∫ (2x)/(x²−1) dx。分母因式分解为 (x−1)(x+1)。设 2x/[(x−1)(x+1)] = A/(x−1) + B/(x+1)，解得 A=1，B=1。积分得 ln |x−1| + ln |x+1| + C = ln |x²−1| + C。

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7. Parametric Integration | 参数方程积分

rea under the curve between parameter values t = α and t = β is given by ∫ y dx = ∫ g(t) f ' seful for loops and curves where y cannot easily be expressed as a function of x directly.

当曲线由参数方程 x = f(t)、y = g(t) 定义时，在参数 t = α 到 t = β 之间的曲线下方面积为 ∫ y dx = ∫ g(t) f '(t) dt，积分限用 t 表示。该方法对于环状曲线以及 y 难以直接表示为 x 的函数的曲线尤其有用。

tegrate from the start of the loop (α) to the point where the curve returns (β), and then double the result if symmetrical. Also, the formula ∫ x dy = ∫ f(t) g'(t) dt can be used if rotating about the y‑axis or calculating areas with respect to y.

需特别留意方向：积分限必须对应沿曲线的运行方向。对于完整环线，通常从环线起点 (α) 积分至回到该点处 (β)，若对称可加倍结果。此外，公式 ∫ x dy = ∫ f(t) g'(t) dt 可在绕 y 轴旋转或计算关于 y 的面积时使用。

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8. Area Between Curves | 曲线间面积

ounded by two curves y = f(x) (top) and y = g(x) (bottom) between x = a and x = b is A = ∫_a^b [f(x) − g(x)] dx. It is essential to identify which function is uppermost over the interval. If the curves cross, split the integral at the intersection point(s) and take the absolute value of each section's area.

由两条曲线 y = f(x)（上方）与 y = g(x)（下方）在 x = a 到 x = b 之间所围成的面积是 A = ∫_a^b [f(x) − g(x)] dx。关键在于确定在区间上哪条曲线在上方。如果曲线相交，应在交点处拆分积分，并对每一段的面积取绝对值。

re expressed as x = f(y) and the region is bounded horizontally. The formula becomes A = ∫_c^d [right − left] dy. In exam questions, sketching the region first is always a wise step to avoid sign errors.

有时对 y 积分更方便，尤其当曲线以 x = f(y) 形式给出且区域为水平边界时。此时公式为 A = ∫_c^d [右曲线 − 左曲线] dy。考试中，先画草图是避免符号错误的好习惯。

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9. Volumes of Revolution | 旋转体体积

When a region is rotated about the x‑axis, the volume of the solid generated is V = π ∫_a^b y² dx. For rotation about the y‑axis, use V = π ∫_c^d x² dy. Both formulas assume the region is swept out completely and that the curve does not cross the axis of rotation within the interval.

将一区域绕 x 轴旋转所产生的立体体积为 V = π ∫_a^b y² dx。绕 y 轴旋转则使用 V = π ∫_c^d x² dy。两个公式都假设区域被完整地旋转，且曲线在区间内不穿过旋转轴。

If the region is bounded by two curves, the volume is given by π ∫ (outer² − inner² or dy). With parametric equations x = f(t), y = g(t), the volume of revolution about the x‑axis becomes V = π ∫_α^β g(t)² f '(t) dt. Remember to square the y‑term before integrating and be careful with the limits.

若区域由两条曲线围成，则体积为 π ∫ (外半径² − 内半径²) dx（或 dy）。对于参数方程 x = f(t), y = g(t)，绕 x 轴旋转的体积公式为 V = π ∫_α^β g(t)² f '(t) dt。注意先平方 y 项再积分，并仔细处理积分限。

Example: The region under y = √x from x = 0 to x = 4 is rotated about the x‑axis. V = π ∫₀⁴ (√x)² dx = π ∫₀⁴ x dx = π [x²/2]₀⁴ = 8π.

例题：将 y = √x 从 x = 0 到 x = 4 下方的区域绕 x 轴旋转。V = π ∫₀⁴ (√x)² dx = π ∫₀⁴ x dx = π [x²/2]₀⁴ = 8π。

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10. The Trapezium Rule | 梯形法则

timate. Divide the interval [a, b] into n equal strips of width h = (b−a)/n. The approximate integral is:

当定积分无法解析求出，或函数仅以表格形式给出时，梯形法则可给出近似值。将区间 [a, b] 等分为 n 个宽度为 h = (b−a)/n 的小条，近似积分公式为：

∫_a^b y dx ≈ ½ h [ (y₀ + y_n) + 2(y₁ + y₂ + … + y_n−1) ]

Increasing the number of strips generally improves accuracy, but the method tends to overestimate when the curve is convex (bending upwards) and underestimate when concave. In exam questions, you may be asked to comment on whether the estimate is an over‑ or underestimate based on the shape of the graph.

增加小条数通常能提高精度，但当曲线为凸（向上弯曲）时该方法容易高估，为凹时则低估。考试中可能要求根据图形形状判断估值是偏高还是偏低。

Apply the formula carefully by listing the ordinates y₀, y₁, …, y_n and then substituting. A table is helpful. Always check the number of ordinates: n+1 ordinates correspond to n strips.

应用公式时，先列出各纵坐标 y₀, y₁, …, y_n 再代入计算，用表格辅助不易出错。注意检查纵坐标的个数：n+1 个纵坐标对应 n 个小条。

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11. Solving Differential Equations | 微分方程求解

In A‑Level, differential equations are typically first‑ of integration. The solution may be left implicitly or rearranged into an explicit form y = … if possible.

A‑Level 中的微分方程通常是一阶可分离变量的。将其整理为 ∫ g(y) dy = ∫ f(x) dx 的形式，两边积分，然后利用给定的初始

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