Integrating Standard Functions | 标准函数积分

📚 Integrating Standard Functions | 标准函数积分

Integration is the reverse process of differentiation. In A-Level Edexcel Mathematics, you are expected to recognise and apply a set of standard integrals immediately, without always deriving them from first principles.

积分是微分的逆运算。在 A-Level Edexcel 数学中，学生需要熟练掌握并直接应用一组标准积分，而不必每次都从第一原理推导。

The general idea is that if F'(x) = f(x), then ∫f(x) dx = F(x) + C, where C is an arbitrary constant of integration.

基本思想是：如果 F'(x) = f(x)，那么 ∫f(x) dx = F(x) + C，其中 C 是任意积分常数。

This article summarises the standard integrals required for the Edexcel A-Level Mathematics course and shows how to use them with confidence.

本文总结了 Edexcel A-Level 数学课程要求的标准积分，并演示如何自信地使用它们。

1. Introduction to Standard Integrals | 标准积分简介

In A-Level Mathematics, a standard integral is one that you should be able to write down immediately, often because it is the exact reverse of a known derivative.

在 A-Level 数学中，标准积分是指你应该能够立刻写出的积分，通常因为这些积分正好是已知导数的逆运算。

For example, since d/dx(sin x) = cos x, it follows that ∫cos x dx = sin x + C. This reverse relationship is the key to building a reliable table of standard integrals.

例如，因为 d/dx(sin x) = cos x，所以 ∫cos x dx = sin x + C。这种逆关系是建立可靠标准积分表的关键。

You will be expected to use these results in pure mathematics problems, including integration by substitution, integration by parts, and when solving differential equations.

在纯数学问题中，包括换元积分法、分部积分法以及求解微分方程时，都需要用到这些结果。

Make sure that you know both the basic forms and the forms involving linear functions of x, such as ax + b.

务必同时掌握基本形式以及含有 x 的线性函数（如 ax + b）的形式。

2. The Power Rule | 幂函数积分法则

For any real number n except -1, the integral of xⁿ is given by:

对于任何实数 n（除 -1 外），xⁿ 的积分为

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