定积分与不定积分解题方法与策略研究

Research on Problem-Solving Methods and Strategies for Definite and Indefinite Integrals

摘要: 积分学作为高等数学的核心支柱之一，定积分与不定积分的求解能力直接决定学习者对微积分理论体系的掌握深度。本文系统梳理积分求解的各类核心方法与进阶技巧，将函数积分分为简单函数、复杂函数和抽象函数三种类型，并构建了“函数类型识别–方法适配选择–问题转化求解”的完整解题框架。研究内容涵盖直接积分法、函数化简技巧、凑微分法、换元积分法、分部积分法及奇偶性在定积分中的优化应用，并深入探讨抽象函数的积分求解策略。通过分类归纳与典型案例剖析，本文旨在为学习者提供一套系统化、可操作的积分求解思维范式，助力提升解题效率与数学逻辑思维能力。

Abstract: As a core pillar of advanced mathematics, proficiency in solving definite and indefinite integrals is a critical determinant of learners’ in-depth comprehension of the theoretical framework of calculus. This paper systematically reviews fundamental approaches and advanced techniques for integral solving, and classifies integrable functions into three categories: elementary functions, complex functions and abstract functions. On this basis, a complete problem-solving paradigm is constructed, which follows the logical chain of “function type identification—adaptive method selection—problem transformation and resolution”. The research scope encompasses direct integration, function simplification, differential matching, substitution method, integration by parts, as well as the optimized application of parity properties in definite integral computation; in addition, it further explores targeted solving strategies for abstract function integrals. Through classification analysis and typical case studies, this paper intends to provide learners with a systematic and operable thinking paradigm for integral solving, thereby improving problem-solving efficiency and cultivating mathematical logical reasoning abilities.