Edexcel A Level Maths Year 2: Integration Techniques | Edexcel A Level 数学第二年：积分技巧

📚 Edexcel A Level Maths Year 2: Integration Techniques | Edexcel A Level 数学第二年：积分技巧

Integration in Edexcel A Level Mathematics Year 2 extends the basic ideas from Year 1 into a much wider toolkit. You will learn how to reverse the chain rule, integrate by substitution, apply integration by parts, split rational functions into partial fractions, solve simple differential equations, and find areas under curves defined in different ways. Mastering these methods is essential for the Pure Mathematics Paper 1 and Paper 2, where integration questions often carry high marks and combine several techniques.

Edexcel A Level 数学第二年的积分内容在 Year 1 基础之上大幅拓展。你将学习逆链式法则、换元积分法、分部积分法、将有理函数拆分为部分分式、解简单微分方程，以及计算不同表达形式曲线下的面积。掌握这些方法对 Pure Mathematics Paper 1 和 Paper 2 至关重要，积分题常常分值较高，并且会把多种技巧综合在一起考查。

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1. Standard Integrals and Reversing Differentiation | 标准积分与微分逆运算

The fundamental idea in Year 2 is that integration is the reverse of differentiation. If you know the derivative of a function, you immediately know an antiderivative. For the power rule, the formula is reversed: to integrate xⁿ, you add one to the power and divide by the new power, provided n is not equal to -1.

第二年最基本的理念是：积分是微分的逆运算。如果知道一个函数的导数，就立刻知道它的一个原函数。对于幂函数法则，积分公式反过来使用：对 xⁿ 积分，把指数加 1，再除以新的指数，条件是 n 不等于 -1。

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, n ≠ -1

The constant of integration C must always be included for indefinite integrals. Omitting the constant loses a mark in Edexcel exam papers. You also need to know standard integrals for exponential, trigonometric, and logarithmic functions, because they appear repeatedly in longer questions.

不定积分必须始终加上积分常数 C。Edexcel 考试中漏写常数会扣分。你还需要熟练掌握指数函数、三角函数和对数函数的标准积分，因为它们在较长的题目中反复出现。

Function	Integral	中文
eˣ	eˣ + C	eˣ 的积分是 eˣ + C
1/x	ln|x| + C	1/x 的积分是 ln|x| + C
sin x	-cos x + C	sin x 的积分是 -cos x + C
cos x	sin x + C	cos x 的积分是 sin x + C
sec² x	tan x + C	sec² x 的积分是 tan x + C

You should be able to write these down instantly. They are not provided in the formula booklet, so secure recall is necessary before attempting harder integration problems.

你应该能够立刻写出这些结果。公式表里不会给出这些标准积分，所以在做更难的积分题之前，必须牢固记忆。

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2. Reverse Chain Rule | 逆链式法则

The reverse chain rule is used when the integrand is the derivative of a function multiplied by a constant. If you recognise that the numerator is the derivative of the denominator, or that the outside function has been differentiated, you can integrate directly by inspection.

当被积函数是一个函数的导数乘以某个常数时，可以使用逆链式法则。如果你发现分子是分母的导数，或者外层函数已经被求导，就能通过观察直接积分。

∫ k f'(x) e^f(x) dx = k e^f(x) + C

For example, ∫ 2x e^x² dx = e^x² ric functions: ∫ cos 3x dx = (1/3) sin 3x + C, since the derivative of sin 3x is 3 cos 3x.

例如 ∫ 2x e^x² dx = e^x² + C，因为 x² 的导数是 2x。同样的方法适用于三角函数：∫ cos 3x dx = (1/3) sin 3x + C，因为 sin 3x 的导数是 3 cos 3x。

grating linear functions inside a function, you can use the reverse chain rule formula ∫ f'(ax+b) dx = (1/a) f(ax+b) + C. This avoids a full substitution and saves time in the exam.

对线性内层函数积分时，可以使用逆链式法则公式 ∫ f'(ax+b) dx = (1/a) f(ax+b) + C。这能避免完整的换元过程，在考试中节省时间。

Always check your answer by differentiating. If the derivative does not return the original integrand, you have missed a constant factor from the chain rule.

始终通过求导来检查答案。如果求导结果不能还原为原被积函数，就说明你遗漏了链式法则产生的常数因子。

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3. Integration by Substitution | 换元积分法

Integration by substitution is the most powerful Year 2 technique for integrals that are not immediately standard. The Edexcel specification often gives you the substitution in the question, but you must also know how to make a sensible choice when it is not provided.

换元积分法是第二年处理非标准积分最强大的方法。Edexcel 考试大纲通常会在题目中给出换元式，但你也需要知道在没有给出时如何做出合理的选择。

To carry out substitution, replace x with a function of u, find dx in terms of du, change the limits if it is a definite integral, and express the whole integral in terms of u only. After integrating, substitute back to the original variable unless the limits were changed.

进行换元时，把 x 表示成 u 的函数，求出 dx 关于 du 的表达式，如果是定积分则更换积分限，并把整个积分只用 u 表示。积分完成后，除非已经更换了积分限，否则要把变量换回原来的字母。

Let u = g(x), then du/dx = g'(x), so dx = du / g'(x)

For example, to evaluate ∫ x√x²+1 dx, let u = x²+1. Then du/dx = 2x, so x dx = (1/2) du. The integral becomes (1/2) ∫ √u du = (1/2) · (2/3) u^3/2 + C = (1/3)(x²+1)^3/2 + C.

例如，计算 ∫ x√x²+1 dx，令 u = x²+1。则 du/dx = 2x，所以 x dx = (1/2) du。积分变为 (1/2) ∫ √u du = (1/2) · (2/3) u^3/2 + C = (1/3)(x²+1)^3/2 + C。

In definite integrals, remember to convert the x-limits into u-limits before evaluating. This avoids having to substitute back to x, and it is a common source of arithmetic errors if forgotten.

对于定积分，记得在计算前把 x 的积分限转换成 u 的积分限。这样可以避免再换回 x，而且忘记换限是常见的计算错误来源。

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4. Integration by Parts | 分部积分法

Integration by parts is the reverse of the product rule. It is used when the integrand is a product of two functions, typically a polynomial multiplied by an exponential, sine, cosine, or logarithm.

分部积分法是乘积法则的逆运算。当被积函数是两个函数的乘积时使用，通常是多项式乘以指数函数、正弦、余弦或对数函数。

∫ u dv/dx dx = uv - ∫ v du/dx dx

Choose u as the function that simplifies when differentiated. The LIATE rule is helpful: Logarithm, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Let u be the earlier category and dv/dx be the later category.

选择 u 时要选求导后会变得更简单的函数。LIATE 法则很有用：对数函数、反三角函数、代数函数、三角函数、指数函数。把较早出现的类别设为 u，把较晚出现的类别设为 dv/dx。

For example, to integrate ∫ x eˣ dx, let u = x and dv/dx = eˣ. Then du/dx = 1 and v = eˣ. The formula gives x eˣ - ∫ eˣ dx = x eˣ - eˣ + C.

例如，计算 ∫ x eˣ dx，令 u = x，dv/dx = eˣ。则 du/dx = 1，v = eˣ。代入公式得到 x eˣ - ∫ eˣ dx = x eˣ - eˣ + C。

, dv/dx = 1. This gives ∫ ln x dx = x ln x - x + C, a standard result you should memorise.

当被积函数只有 ln x 时，把它写成 1 × ln x，并令 u = ln x，dv/dx = 1。这样得到 ∫ ln x dx = x ln x - x + C，这是一个需要记住的标准结果。

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5. Partial Fractions and Rational Integration | 部分分式与有理函数积分

Many rational functions must be split into partial fractions before they can be integrated. The Edexcel Year 2 course covers linear factors and repeated linear factors. For example, a fraction with denominator (x+1)(x-2) can be written as A/(x+1) + B/(x-2).

许多有理函数在积分前需要拆分为部分分式。Edexcel 第二年课程涵盖线性因式和重复线性因式。例如，分母为 (x+1)(x-2) 的分式可以写成 A/(x+1) + B/(x-2)。

of x. Once split, each term integrates to a natural logarithm, such as ∫ A/(x+1) dx = A ln|x+1| + C.

求常数 A 和 B 时，先乘以公分母，再比较系数或代入方便的 x 值。拆分后，每一项都能积分为自然对数，例如 ∫ A/(x+1) dx = A ln|x+1| + C。

∫ (2x+1)/((x+1)(x-2)) dx = ∫ (1/(x+1) + 1/(x-2)) dx

This gives ln|x+1| + ln|x-2| + C. If the numerator has degree equal to or greater than the denominator, you must first divide the polynomial to get a proper fraction. This is a common exam trap.

结果得到 ln|x+1| + ln|x-2| + C。如果分子的次数大于或等于分母的次数，必须先进行多项式除法，把分式化为真分式。这是考试中常见的陷阱。

Repeated roots such as (x-2)² require a slightly different decomposition: A/(x-2) + B/(x-2)². The second term integrates to -B/(x-2) using the power rule, not to a logarithm.

形如 (x-2)² 的重根需要稍有不同的拆分：A/(x-2) + B/(x-2)²。第二项使用幂法则积分得到 -B/(x-2)，而不是对数函数。

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6. Integrating Trigonometric Functions | 三角函数的积分

Year 2 requires fluency with powers and products of trigonometric functions. You will often use trigonometric identities to rewrite the integrand into a standard form before integrating.

第二年要求熟练处理三角函数的高次幂和乘积。你通常需要先用三角恒等式把被积函数改写成标准形式，再进行积分。

s² x = (1 + cos 2x)/2. For products like sin mx cos nx, use the product-to-sum identities.

对于正弦和余弦的偶次幂，使用二倍角恒等式：sin² x = (1 - cos 2x)/2，cos² x = (1 + cos 2x)/2。对于 sin mx cos nx 这类乘积，使用积化和差恒等式。

∫ sin² x dx = ∫ (1 - cos 2x)/2 dx = x/2 - sin 2x/4 + C

itution u = cos x. This converts the integral into a polynomial that is easy to integrate.

对于 sin³ x 这样的奇次幂，分离出一个因子：sin³ x = sin x (1 - cos² x)，然后令 u = cos x 进行换元。这样就把积分转化为容易处理的多项式。

You must also know integrals of tan x and sec x. Recall that ∫ tan x dx = ln|sec x| + C and ∫ sec x dx = ln|sec x + tan x| + C. These can appear in harder differential equation contexts.

你还需要知道 tan x 和 sec x 的积分。记住 ∫ tan x dx = ln|sec x| + C 和 ∫ sec x dx = ln|sec x + tan x| + C。它们可能出现在较难的微分方程情境中。

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7. Differential Equations by Separation of Variables | 变量分离法解微分方程

ration of variables: move all y-terms to the left with dy, all x-terms to the right with dx, then integrate both sides.

第二年要求解形如 dy/dx = f(x)g(y) 的一阶微分方程。这种方法称为变量分离法：把所有含 y 的项与 dy 移到左边，所有含 x 的项与 dx 移到右边，然后对两边积分。

∫ (1/g(y)) dy = ∫ f(x) dx

hen x = 0, substitute these values to find the constant and write the particular solution.

积分后会得到含有任意常数的通解。如果给出初始条件，例如 x = 0 时 y = 2，代入这些值求出常数，就能写出特解。

For example, solve dy/dx = 2xy. Separate to get ∫ (1/y) dy = ∫ 2x dx, so ln|y| = x² + C. Exponentiating gives y = A e^x², where A = ±e^C. The constant can be found using an initial condition.

例如，解微分方程 dy/dx = 2xy。分离变量得到 ∫ (1/y) dy = ∫ 2x dx，所以 ln|y| = x² + C。取指数得到 y = A e^x²，其中 A = ±e^C。可使用初始条件确定常数。

. You must be able to rearrange the final answer into the required form, such as y = f(x).

Edexcel 经常要求结合情境解释解，例如人口增长、冷却或放射性衰变。你必须能把最终答案整理成题目要求的形式，例如 y = f(x)。

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8. Area Under a Curve and Parametric Integration | 曲线下面积与参数积分

he area under the curve between the parameter values t₁ and t₂ is given by ∫ y (dx/dt) dt.

第二年将面积计算拓展到参数方程定义的曲线。如果 x = x(t)，y = y(t)，那么参数 t 从 t₁ 到 t₂ 之间曲线下的面积为 ∫ y (dx/dt) dt。

Area = ∫ y dx = ∫ y(t) (dx/dt) dt

This is useful for finding areas enclosed by parametric curves such as circles, ellipses, or cycloids. You must be careful to integrate with respect to the parameter t and use the correct limits from the question.

这对于求圆、椭圆或摆线等参数曲线围成的面积非常有用。必须注意对参数 t 积分，并使用题目给出的正确积分限。

When finding the area between a curve and the x-axis, remember that regions below the x-axis give a negative signed area. To find the total physical area, split the integral or take absolute value where the curve crosses the axis.

求曲线与 x 轴之间的面积时，记住 x 轴下方的区域会给出负的符号面积。要计算总实际面积，必须在曲线穿过 x 轴的位置拆分积分或取绝对值。

f(x), you should also be comfortable with definite integrals and areas between two curves: subtract the lower function from the upper function before integrating.

对于直角坐标形式 y = f(x) 的曲线，你还需要熟练掌握定积分和两条曲线之间的面积：积分前用上面的函数减去下面的函数。

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9. Trapezium Rule and Numerical Integration | 梯形法则与数值积分

The trapezium rule is used to approximate the value of a definite integral when the function cannot be integrated analytically. It is especially important for functions defined by experimental data or complicated expressions.

当函数无法用解析方法积分时，梯形法则用于近似计算定积分的值。对于由实验数据定义的函数或复杂表达式，它尤其重要。

∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + ... + yₙ₋₁)]

d yᵢ are the function values at equally spaced x-values. The more strips you use, the smaller the strip width and the better the approximation.

这里 h = (b - a)/n 是每个小区间的宽度，yᵢ 是等距 x 值处的函数值。使用的区间越多，宽度越小，近似值就越精确。

Edexcel questions may ask you to use the trapezium rule with a given number of ordinates, or to state whether the approximation is an overestimate or underestimate. This depends on the shape of the curve: if the curve is concave up, the trapezium rule overestimates; if concave down, it underestimates.

Edexcel 题目可能要求用给定的纵坐标数量使用梯形法则，或判断近似值是偏大还是偏小。这取决于曲线的形状：如果曲线是凹向上的，梯形法则会偏大；如果凹向下，则会偏小。

You must set out your working clearly with a table of y-values. Marks are awarded for the table, the correct h value, and the substitution into the trapezium rule formula.

你必须清楚地列出 y 值表。分数会给在表格、正确的 h 值以及代入梯形法则公式的过程上。

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10. Common Mistakes and Exam Tips | 常见错误与备考建议

Integration is one of the most heavily weighted topics in Edexcel A Level Maths Year 2. Even small slips can cost several marks, so it is essential to practise under timed conditions and check every answer by differentiating.

积分是 Edexcel A Level 数学第二年权重最高的主题之一。即使很小的失误也可能丢掉好几分，因此必须在计时条件下练习，并通过求导检查每一个答案。

The most common mistake is forgetting the constant of integration in an indefinite integral. Another is miscopying the chain rule factor when using substitution or reverse chain rule. Always check that differentiating your answer returns the original integrand exactly.

最常见的错误是在不定积分中忘记积分常数。另一个错误是在换元或逆链式法则中抄错链式法则因子。务必检查你的答案求导后能否完全还原为原被积函数。

In definite integrals, convert the limits immediately after substitution. Failing to do so means you may accidentally keep x-limits with a u-integral, giving an incorrect numerical answer.

在定积分中，换元后要立刻转换积分限。如果不这样做，可能会把 x 的积分限与 u 的积分混用，导致数值答案错误。

Finally, read the question carefully: some questions ask for an exact value, others for an approximation or a particular solution. Show all integration steps, label constants clearly, and write your final answer in the requested form.

最后，仔细审题：有些问题要求精确值，有些要求近似值或特解。展示所有积分步骤，清楚标明常数，并按照题目要求的形式书写最终答案。

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