Integration Techniques for Edexcel A Level Pure Maths | 爱德思 A-Level 纯数学积分技巧

📚 Integration Techniques for Edexcel A Level Pure Maths | 爱德思 A-Level 纯数学积分技巧

Integration is one of the largest and most interconnected topics in Edexcel A Level Mathematics. It builds directly on differentiation, algebra, trigonometry and coordinate geometry, and it appears in both Pure Mathematics papers and applied contexts such as kinematics and volumes. Students need to recognise which integration strategy fits a given expression and carry it through accurately under timed conditions.

积分是爱德思 A-Level 数学中最大、关联性最强的主题之一。它直接建立在微分、代数、三角学和坐标几何的基础上，既出现在纯数学试卷中，也出现在运动学和旋转体体积等应用情境中。学生需要能识别哪一种积分策略适合给定的表达式，并在限时条件下准确完成计算。

1. Standard Integrals and the Constant of Integration | 标准积分与积分常数

Before attempting advanced methods, you must know the standard results instantly. For any real constant c, the indefinite integral of xⁿ is xⁿ⁺¹/(n+1) + c, provided n ≠ −1. The constant c is essential because differentiation removes constants, so infinitely many functions share the same derivative.

在尝试更高级的方法之前，你必须能立刻写出标准结果。对任意实常数 c，xⁿ 的不定积分是 xⁿ⁺¹/(n+1) + c，其中 n ≠ −1。常数 c 必不可少，因为微分会消去常数，所以无穷多个函数拥有相同的导数。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, n ≠ −1

When n = −1, the result is ln|x| + c. Other core results include ∫ eˣ dx = eˣ + c, ∫ sin x dx = −cos x + c and ∫ cos x dx = sin x + c. You should also know ∫ sec²x dx = tan x + c and ∫ 1/(1+x²) dx = arctan x + c.

当 n = −1 时，结果是 ln|x| + c。其他核心结果包括 ∫ eˣ dx = eˣ + c，∫ sin x dx = −cos x + c 以及 ∫ cos x dx = sin x + c。你还应掌握 ∫ sec²x dx = tan x + c 以及 ∫ 1/(1+x²) dx = arctan x + c。

A quick mental check is to differentiate your answer: if the derivative returns to the original integrand, the integral is correct. This habit also helps you remember whether signs and coefficients are right.

一个快速的检验方法是对答案求导：如果求导后回到原来的被积函数，就说明积分正确。这个习惯也有助于你记住符号和系数是否正确。

2. Integrating f(ax+b) | 线性函数代换积分

Edexcel papers often require integrating expressions such as e³ˣ, sin(2x+1) or (5x−2)⁴. These are examples of f(ax+b), where a linear change of variable has been made inside a standard function. If ∫ f(x) dx = F(x) + c, then ∫ f(ax+b) dx = (1/a)F(ax+b) + c.

爱德思试卷经常要求对 e³ˣ、sin(2x+1) 或 (5x−2)⁴ 这样的表达式积分。这些都是 f(ax+b) 的例子，即在标准函数内部进行了线性变量替换。如果 ∫ f(x) dx = F(x) + c，那么 ∫ f(ax+b) dx = (1/a)F(ax+b) + c。

∫ f(ax+b) dx = (1/a)F(ax+b) + c

For example, ∫ cos(3x) dx = (1/3)sin(3x) + c and ∫ e²ˣ dx = (1/2)e²ˣ + c. Always divide by the coefficient of x, not multiply. A common error is writing ∫ e²ˣ dx = 2e²ˣ + c, which is wrong because differentiation of 2e²ˣ gives 4e²ˣ.

例如，∫ cos(3x) dx = (1/3)sin(3x) + c，∫ e²ˣ dx = (1/2)e²ˣ + c。务必除以 x 的系数，而不是乘以它。一个常见错误是把 ∫ e²ˣ dx 写成 2e²ˣ + c，这是错误的，因为对 2e²ˣ 求导得到的是 4e²ˣ。

In definite integrals, apply the same rule before substituting limits. For ∫ from 0 to π/6 of cos(3x) dx, first integrate to (1/3)sin(3x), then substitute the limits to obtain (1/3)sin(π/2) − (1/3)sin 0 = 1/3.

定积分在代入上下限之前也要使用同样的法则。对 ∫ 从 0 到 π/6 的 cos(3x) dx，先积成 (1/3)sin(3x)，再代入上下限，得到 (1/3)sin(π/2) − (1/3)sin 0 = 1/3。

3. Reverse Chain Rule | 逆链式法则

The reverse chain rule handles integrals of the form ∫ f'(g(x))g'(x) dx, where the derivative of the inner function is already present as a factor. In general, if you differentiate f(g(x)) you get f'(g(x))g'(x), so the integral returns to f(g(x)) + c.

逆链式法则处理形如 ∫ f'(g(x))g'(x) dx 的积分，其中内层函数的导数已经作为一个因式出现。一般来说，如果对 f(g(x)) 求导，你会得到 f'(g(x))g'(x)，因此积分回到 f(g(x)) + c。

∫ f'(g(x))g'(x) dx = f(g(x)) + c

A typical example is ∫ 2x e^(x²) dx. Here g(x) = x² and g'(x) = 2x, so the integral is e^(x²) + c. Another is ∫ cos(sin x) cos x dx = sin(sin x) + c if the inner function is sin x and its derivative cos x is present.

一个典型例子是 ∫ 2x e^(x²) dx。这里 g(x) = x²，g'(x) = 2x，所以积分是 e^(x²) + c。另一个例子是 ∫ cos(sin x) cos x dx = sin(sin x) + c，其中内层函数是 sin x，它的导数 cos x 已存在。

You can also use the reverse chain rule with powers: ∫ (3x²+2)(x³+2x)⁵ dx = (1/6)(x³+2x)⁶ + c because the derivative of x³+2x is 3x²+2, and the outside function is the power operation.

你也可以对幂函数使用逆链式法则：∫ (3x²+2)(x³+2x)⁵ dx = (1/6)(x³+2x)⁶ + c，因为 x³+2x 的导数是 3x²+2，而外层函数是幂运算。

4. Integration by Substitution | 换元积分法

When the reverse chain rule is not obvious, choose a substitution u = g(x), differentiate to find du/dx, and replace all x terms. The aim is to reduce the integral to a standard form in u. After integrating with respect to u, substitute back to express the answer in x.

当逆链式法则不明显时，可以选择换元 u = g(x)，求导得到 du/dx，并把所有含 x 的项替换掉。目标是把积分化为关于 u 的标准形式。对 u 积分后，再代回原变量，用 x 表示答案。

∫ f(x) dx = ∫ f(g⁻¹(u)) (dx/du) du

For definite integrals, remember to change the limits. If the original limits are x = a and x = b, they become u = g(a) and u = g(b). You can then evaluate the u-integral directly without substituting back.

对于定积分，一定要改变上下限。如果原上下限是 x = a 和 x = b，它们就变为 u = g(a) 和 u = g(b)。然后你可以直接计算 u 的定积分，而不必代回原变量。

Example: ∫ x√(x

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