关于不定积分的一题多解问题

On Multiple Solutions of Indefinite Integral Problems

不定积分在大学数学中有着举足轻重的地位,是学习大学物理、概率论与数理统计、微分方程等课程的基础知识.求解不定积分的技巧性很强,而且目前可用的工具也很少,只有换元积分法和分部积分法两种最基本的解法.从3道不定积分例题着手,采用换元积分法、分部积分法、添项法、万能公式法、欧拉代换法等多种方法对3道例题进行求解,最终总结了含根号或者幂次方、含分式、含三角函数这3类不定积分题型的求解方法.通过一题多解的方式,帮助学生掌握系统的知识,培养发散性思维,解决更多类型的不定积分的求解方法.

Indefinite integral plays an important role in college mathematics,which is the basic knowledge of college physics,probability theory and mathematical statistics,differential equation and other courses.The technique of solving indefinite integral is strong,and there are few tools available.For the indefinite integral problems,there are only the two most basic solutions.They are the most basic substitution integration method and integration by parts of.This paper starts from three indefinite integral problems that are not easy to be solved.By using substitution integral method,integration by parts method,addition method,universal formula method,Euler substitution method and other methods,three examples are solved.Finally the method of indefinite integrals with square root or power,with fractions,with trigonometric functions are summarized.Through a way of multiple solutions to help student master knowledge systematically and foster divergent thinking.Do three by analogy,solve more types of indefinite integral problems.