商的分部积分法及其应用

Division Integral Method of Quotient and Its Application

从某种意义上来讲,求函数的不定积分是一种运算,牛顿–莱布尼茨公式(微积分基本定理)揭示了定积分与不定积分的联系,为解决定积分以及多重积分乃至曲线积分和曲面积分问题提供了极大的方便,因而求函数的不定积分在积分学习中占有极其重要的地位。求不定积分运算可视为求导的逆运算,因而在现有市面教材中由函数和、差、积的求导法则相应推出了函数和、差、积(分部积分公式)的积分法则,而唯独没有商的积分法则。本文通过函数商的求导法则推导出函数商的积分法则,我们称之为商的分部积分法则。通过历年全国硕士研究生入学统一考试试题和广东省大学生数学竞赛试题作为例子说明商的分部积分法则的应用。该法则为解决被积函数中有分式的积分提供了一种思路。

In a sense, the indefinite integral of a function is an operation. Newton-Leibniz formula (The Fun-damental Theorem of Calculus) reveals the relationship between definite integral and indefinite integral, which provides great convenience for solving definite integral, multiple integral, and even curve integral and surface integral. Therefore, the indefinite integral of a function occupies an extremely important position in integral learning. The indefinite integral operation can be re-garded as an inverse operation for derivation. Therefore, in the existing market teaching materials, the integral laws of the sum, difference and integral (partial integral formula) of indefinite integral are derived from the derivative laws of function sum, difference and integral, but there is only no integral law of quotient. In this paper, the integral law of function quotient is derived from the derivative law of function quotient, which is called the quotient partial integral law. The application of the quotient’ s partial integral rule is illustrated by examples of the national unified entrance examination for postgraduate students and the mathematical competition test for college students in Guangdong Province over the years. This rule provides a way of thinking for solving the integral of fraction in the integrand function.