第一类换元积分法的学习难点、题型归类与教学策略研究

Learning Difficulties, Problem Classification, and Teaching Strategies for the First Integration by Substitution

摘要: 第一类换元积分法是积分运算的基础方法。该方法通过引入中间变量，将复杂积分转化为适配基本积分公式的标准形式，以此简化积分运算。本文从第一类换元积分法的基本原理入手，系统梳理其解题步骤，深入剖析学生学习过程中的难点及其成因，对典型题型进行分类归纳，并补充完善基本积分公式表。基于上述研究内容，本文结合学习难点与题型分类思路，提出针对性教学策略。研究表明：清晰的题型划分、针对学习难点的专项教学干预，能够有效改善课堂教学效果，帮助学生理解并掌握该方法蕴含的数学思想，拓展数学思维，提升解题能力。本文研究成果也可为高等数学课程教学相关研究提供理论依据与实践参考。

Abstract: The first integration by substitution is a fundamental method in integral calculus. By introducing an intermediate variable, this method transforms complex integrals into forms consistent with basic integral formulas, thereby simplifying the integration process. This paper begins with the fundamental principle of the first integration by substitution, systematically outlines its solution steps, thoroughly analyzes the difficulties encountered by students in learning and their underlying causes, classifies typical problem types, and supplements the list of basic integral formulas accordingly. Based on the above analysis, the paper proposes targeted teaching strategies that integrate the identified learning difficulties with the classification of problem types. The findings indicate that clear classification of problem types and targeted instructional interventions addressing learning difficulties can effectively improve classroom teaching outcomes, help students understand the underlying mathematical ideas of the method, broaden their mathematical thinking, and enhance their problem-solving abilities. The results of this study may also serve as a theoretical foundation and practical reference for research on the teaching of advanced mathematics courses.