Mixed Exercise 13: Integration | 混合练习13：积分

📚 Mixed Exercise 13: Integration | 混合练习13：积分

Mixed Exercise 13 is the end-of-chapter mixed review for Chapter 13 of the Edexcel AS Pure Mathematics Year 1 course. It brings together indefinite integration, definite integration, finding the constant of integration, and area problems under curves and between curves.

混合练习13是Edexcel AS纯数学一年级第13章的章末综合练习。它综合考查不定积分、定积分、积分常数的求解，以及曲线下方和曲线之间的面积问题。

1. Overview of Mixed Exercise 13 | 混合练习13 总览

This exercise is designed to test every skill from the integration chapter in one mixed set. You will see polynomials, powers of x, definite integrals with numerical limits, curves defined through their derivatives, and area problems that require you to choose the correct limits and subtract the correct functions.

本练习旨在用一组混合题考查积分章节的全部技能。你会遇到多项式、x 的幂、带数值上下限的定积分、由导数定义的曲线，以及需要你正确选择上下限并正确相减的面积问题。

In Edexcel AS Mathematics, integration is treated as the reverse process of differentiation. Mastery of the basic rule for xⁿ and careful use of the constant of integration are the two most important foundations.

在Edexcel AS数学中，积分被看作微分的逆运算。熟练掌握 xⁿ 的基本积分公式，并正确使用积分常数，是两个最重要的基础。

2. The Integration Toolkit | 积分工具包

The heart of this exercise is reversing differentiation. If dy/dx = xⁿ, then y = xⁿ⁺¹/(n + 1) + c, provided n ≠ -1. The constant c appears because differentiation of any constant gives zero.

本练习的核心是微分的逆运算。如果 dy/dx = xⁿ，那么 y = xⁿ⁺¹/(n + 1) + c，前提是 n ≠ -1。常数 c 出现的原因是任何常数求导后都为零。

Constants are integrated as kx, and sums or differences are integrated term by term. Never try to integrate a product by integrating each factor separately unless it has been expanded into a sum first.

常数项积分后为 kx，和式或差式可逐项积分。除非先把乘积展开为和式，否则不要通过分别积分每个因式来积分乘积。

At AS level, do not use the power rule when n = -1, because that would give division by zero. The integral of x⁻¹ is covered later in the A-level course.

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