微分思想在高等数学教学中的运用——不定积分教学设计
Ideas of Differential Used in the Teaching of Advanced Mathematics—Teaching Design for Indefinite Integral

在微积分的教学中，换元积分部分的内容往往是其中的难点。积分是微分的逆运算，而不管是第一类换元积分还是第二类换元积分，都与复合函数微分形式不变性有紧密关系。学习换元积分的困难往往是由于对微分理解不够透彻。微分和不定积分，是数学分析/高等数学/微积分中的重要概念，两者有着紧密联系。在微积分的教学中，应当强调微分思想，引导学生通过微分来理解链式法则，进而掌握换元积分的原理。在笔者的教学实践中发现，通过微分，能够更好地理解不定积分中相关的原理，如复合函数求导的链式法则，第一类及第二类换元积分，以及分部积分。本文介绍通过微分思想来理解不定积分的思路，并讨论在不定积分教学中如何结合微分的思想，提高教学效果。
In the teaching of calculus, the integration by substitution is usually the difficult point. Integration is the inverse operation of differential, and both the substitution of the first kind and the second kind are closely related to the invariance of differential form of the composite function. The difficulty in learning an integration by substitution is often caused by the incomplete understanding of differential. Differential and indefinite integral are important concepts in mathematical analysis/higher mathematics/calculus, and they are closely related. In the teaching of calculus, the idea of differential should be emphasized, so as to guide students to understand the chain rule through differential, and thus understand the principles of integration by substitution. In authors’ teaching practices, it is found that, the related principles of the indefinite integral can be better understood via the idea of differential, such as the chain rule of the composite function in differentiation, the integration by substitution of the first and the second kind, and integration by parts. In this paper, it is demonstrated that the principles of differentiation and indefinite integral can be understood based on the idea of differential, and how the teaching efficiency can be improved if the teaching of indefinite integral is combined with the idea of differentials.