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“数学通报”1962年第8期上登載了张广柱和帅启慧两位同志合写的“关于定积分換元法则中的若干問题”一文(以下簡称原文),經过初步閱讀后,我感到有必要弄清楚定积分换元法则的适用范围的問題。对这个問題,我不作全面詳尽的探討,只談談自己的一些粗浅看法,借此与张、帅二位同志商榷。原文首先在§1中对于积分簡单地列出了它的換元法则。即假設:1.f(x)是区間[a,b]上的連續函数;2.命x=φ(t),使函数φ(t)合于下列諸条件: (ⅰ) φ(t)在某一区間[α,β]上确定且連續,并且当t在区間[α,β]上变化时,φ(t)的值不超出区間[a,b]的范围(可能发生这样的事情:函数f(x)在比[a,b]更大的区間[A,B]上确定且連續,于是只需要
The “Mathematical Bulletin” No. 8 of 1962 published the article “About Some Problems in the Law of Determining the Integral Eliminating Points” written by the two comrades Zhang Guangzhu and Shuai Qihui (hereinafter referred to as the original). After a preliminary reading, I felt that there were It is necessary to clarify the question of the scope of application of the law of definite integral replacement. Regarding this issue, I will not discuss it in detail. I will only talk about some of my superficial views, so as to discuss with Zhang and Shuai. The original text first briefly listed the law of its substitutions for integration in § 1. That is, assume that: 1.f(x) is a continuous function on the interval [a,b]; 2. Life x=φ(t) makes the function φ(t) fit into the following conditions: (i) φ(t) It is determined and continuous over some interval [α,β], and when t changes in the interval [α,β], the value of φ(t) does not exceed the range of interval [a,b] (this may happen : The function f(x) is determined and continuous over an interval [A,B] larger than [a,b], so only need