📚 Integration Techniques (Edexcel A-Level, pdfjoiner_4-071) | 积分方法（Edexcel A-Level，pdfjoiner_4-071）
pdfjoiner_(4)-071. It covers core techniques required for exam success, from basic antiderivatives to advanced applications such as parametric integration and differential equations.
本文整合了Edexcel A-Level纯数学课程中的关键积分方法，内容源自pdfjoiner_(4)-071所汇编的复习资料。文章涵盖了从基本反导数到参数积分、微分方程等高级应用的核心技巧，旨在帮助考生在考试中取得优异成绩。
1. Introduction to Integration | 积分简介
gral includes an arbitrary constant C because differentiation discards constant terms. The antiderivative is not unique; any function differing only by a constant shares the same derivative.
积分是微分的逆运算。如果 dy/dx = f(x)，则 y = ∫ f(x) dx。不定积分包含任意常数 C，因为微分时常数项会消失。原函数并非唯一；任何仅相差常数的函数具有相同的导数。
In Edexcel A-Level exams, you will need to integrate a variety of functions and apply integration to find areas, volumes, and solutions to differential equations. A solid understanding of standard integrals and algebraic manipulation is essential.
在Edexcel A-Level考试中，你需要对各种函数进行积分，并应用积分求解面积、体积和微分方程。因此，扎实掌握标准积分公式和代数变形技巧至关重要。
2. Standard Integrals and Basic Rules | 标准积分与基本法则
a is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, valid for n ≠ -1. For n = -1, the integral is ∫ 1/x dx = ln|x| + C. Exponential and trigonometric standard integrals are equally common: ∫ eˣ dx = eˣ + C; ∫ sin x dx = -cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C.
最基本的公式是 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，当 n ≠ -1 时成立。对于 n = -1，积分为 ∫ 1/x dx = ln|x| + C。指数函数和三角函数的标准积分同样常见：∫ eˣ dx = eˣ + C；∫ sin x dx = -cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C。
e a, b are constants. This allows term-by-term integration of polynomials and series. A handy summary table is shown below:
积分的线性法则指出 ∫ [af(x) ± bg(x)] dx = a∫ f(x) dx ± b∫ g(x) dx，其中 a、b 为常数。利用该法则可以逐项对多项式和级数进行积分。以下是一个便捷的总结表：
| Function f(x) | 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 |
3. Integration by Substitution | 换元积分法
The substitution rule reverses the chain rule. For an integral of the form ∫ f(g(x)) g'(x) dx, substitute u = g(x) so that du = g') dx, giving ∫ f(u) du. This often transforms a complicated integral into a standard form.
换元积分法源于链式法则的逆操作。对于形如 ∫ f(g(x)) g'(x) dx 的积分，令 u = g(x)，则 du = g'(x) dx，转换为 ∫ f(u) du。这通常能将复杂的积分化为标准形式。
ges from g(a) to g(b). Alternatively, revert to the original variable after integration. For example, evaluate ∫₀¹ 2x ex² . Let u = x², du = 2x dx; when x = 0, u = 0; when x = 1, u = 1. The integral becomes ∫₀¹ eu du = e - 1.
用换元法计算定积分时，必须将积分限转换到新变量 u。如果 x 从 a 到 b，则 u 从 g(a) 到 g(b)。也可以在积分后代回原变量。例如，计算 ∫₀¹ 2x ex² dx：令 u = x²，则 du = 2x dx；当 x = 0 时 u = 0，x = 1 时 u = 1，积分变为 ∫₀¹ eu du = e - 1。
Another classic
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