两类“换元积分法”的联系与区别

On the Relationship and Distinction between Two Types of "Integration by Substitution"

不定积分是高等数学中的教学重点与难点,不定积分计算方法一般被分为换元积分法、直接积分法与分部积分法几种方式,其中,换元积分法又可以分为第一类换元法与第二类换元法两种,帮助学生掌握好第一类积分法与第二类积分法在归类上的联系与区别,能够有效提高学生应用积分法求解积分问题的能力,第一类积分法与第二类积分法最大的区别就是,第一类积分法不需要设置变量,可以通过凑微法和转化法进行计算,而在使用第二类积分法时,就必须要选择好变量进行替换。

Indefinite integral is a key and difficult point of higher mathematics, and the computing methods of indefinite integral are generally classified into integration by substitution, immediate integration and integration by parts, among which integration by substitution can be classified into the first type of substitution and the second type of substitution. To help students master the rela- tionship and distinction between the two types of substitution can effectively improve students＇ ability of using integration methods to solve integration problems. The biggest distinction between the two types of substitution is that variables are not needed in the former but improvising differentiation and conversion method can be used in the computing, while a certain variable must be se- lected to be substituted in the latter.