Discussion Overview
The discussion revolves around the possibility of defining two functions, ##f(x)## and ##g(x)##, such that the integral ##∫f(x)g'(x)dx## does not yield an indefinite solution, while the integral ##∫f'(x)g(x)dx## does yield an indefinite solution. The scope includes mathematical reasoning and exploration of integral properties.
Discussion Character
- Exploratory, Mathematical reasoning, Debate/contested
Main Points Raised
- One participant suggests using the partial integral rule, indicating that if the left side integral is definite, the right side must also be definite, implying that the desired condition may not be achievable.
- Another participant seeks clarification on what is meant by "not valid," questioning if the absolute value function as ##g(x)## is a suitable example due to the non-existence of its derivative at zero.
- A later reply clarifies that "not valid" refers to the integral not yielding results in terms of standard mathematical functions.
- Another participant proposes that if the integral is Riemann, then defining ##g(x)=|x|## could work under certain conditions, specifically if zero is within the integration range.
- There is a suggestion that the derivatives of the functions are well-defined in the integration range, but the integrals themselves may have issues, particularly regarding closed forms.
Areas of Agreement / Disagreement
Participants express differing views on the feasibility of the original question, with some suggesting potential functions while others challenge the assumptions or definitions involved. The discussion remains unresolved regarding the specific functions that meet the criteria.
Contextual Notes
There are limitations regarding the definitions of validity in the context of integrals, as well as the implications of using Riemann versus Lebesgue integration. The discussion does not resolve these aspects.