高等数学中积分上限函数的性质与应用

The Properties and Applications of Integral Upper Limit Functions in Advanced Mathematics

摘要: 积分上限函数是高等数学课程中非常重要的一类函数，由此可以证明微积分基本定理。积分上限函数在各类数学竞赛和考研试题中扮演着重要角色，是常考常新的重要知识点。由于积分上限函数的定义方式不同于普通函数，学生不太容易掌握，是课程学习的一个难点。本文在高等数学教材的基础上，总结归纳了积分上限函数推广的求导公式，并且进一步讨论了周期性和奇偶性。接着本文探讨了积分上限函数的应用，包括证明柯西–施瓦茨不等式，中值问题，以及与微分方程相结合的问题等。

Abstract: The upper limit function of integration is a very important type of function in advanced mathematics, which can prove the basic theorem of calculus. The upper limit function of integration plays an important role in various mathematics competitions and postgraduate entrance exams, and it is often tested and updated. Due to the different definition of integral upper limit functions from common functions, it is not easy for students to master, which is a difficult point in course learning. On the basis of advanced mathematics, this article summarizes the derivation formula of integral upper limit functions, and discusses its periodicity and odevity. Next, this article explores the applications of integral upper limit functions, including proving the Cauchy-Schwartz inequality, mean value problems, and applications related to differential equations.