Since I was a college student, I was told there were many functions that cannot be integrated as simple functions.(I'll give the definition of simple functions at the end of the article). As a TA for calculus now, I've been asked for integrated various functions, certainly, most of them are integrable(in the sense of simple functions). However, how could I know that certain functions are not integrable not merely because I cannot integrate them. (There was one time that one integration on the question sheet daunted all the TAs I asked).
Does anyone know THEORETICAL REASONS why certain functions cannot be integrated as simple funtions? Or could you refer to certain reference containing such materials? Or, could you show me by example, certain "good" function actually don't have "good" integration, I think one famous example could be "$\frac{\sin x}{x}$".
Simple functions:
The functions which is the summations(subtractions),multiplications(divisions), and compositions of the following functions(as well as the functions generated by these operations): $x, \sin x , \cos x, \log x, \sqrt[n]{x}, e^x$.
Answer
A search for "integration in finite terms" will get you many useful results. This paper by Rosenlicht is a very good place to start.
Bibliographic details: Maxwell Rosenlicht, Integration in Finite Terms, American Mathematical Monthly 79 (1972) 963-972.
No comments:
Post a Comment