不定积分换元法的使用条件及推广

Application Conditions and Popularization of the Indefinite Integral Element Replacement Method

摘要: 不定积分换元法是最为常用的积分方法，本文讨论了换元法使用的充分条件，针对第一类和第二类换元法，描述了不满足条件可能引起的错误。有些积分采用常规的换元法比较麻烦，甚至无法求解，本文受定积分几何应用求参数方程给出的曲线弧长方法以及双曲函数恒等变换的启发，增加了换元方法，如双曲代换、参数方程及欧拉代换等换元方法，从而解决了一类带有根式和由隐函数方程确定函数的不定积分，推广了一些函数的不定积分换元方法。

Abstract: The indefinite integral substitution method is the most commonly used integration method. This paper discusses the sufficient conditions for the use of the substitution method, and describes the possible errors caused by not meeting the conditions for the first and second kinds of substitution methods. Some integrals are troublesome and even can’t be solved by using conventional substi-tution methods. Inspired by the method of curve arc length given by the application of parametric equation in definite integral geometry and hyperbolic function identity transformation, this paper adds substitution methods, such as hyperbolic substitution, parametric equation and Euler sub-stitution, so as to solve a class of indefinite integrals with roots and functions determined by implicit function equations. In this paper, the method of indefinite integral transformation of some functions is generalized.