Talk:Simplest integral
This article was based only on an error. The article said:
"The simplest integral of a function is the general integral of that function, with the arbitary constant set equal to 0. Or, more formally, it is the unique integral of a function which has no constant [[terms in a function|terms]] in it."
So if the "integral" is sec2(x) is that the "simplest integral" because it has no constant term, even though it is the same as tan2(x)+1, which does have a constant term? Or is tan2(x) the "simplest integral", having no constant term, so that tan2(x)+1 would not be the "simplest integral", even though it is the same as sec2(x), which appears to have no constant term? That example is typical. No one antiderivative of a function is privileged as "simpler" than all others. -- Mike Hardy
What I meant was, if we have sec2 and sec2 + 1, the sec2 is easier, even though they are both general integrals of the function. This concept is useful in, for example, the integrating factor method, and definite integration. We ignore the arbitrary constant, that is, we set it to 0, and the sums are simplified. If you can suggest a better name for this integral, your suggestion would be welcome. -- Adam Burley (Kidburla2002)
(note: in all the books I have read, it is called the simplest integral)
But sec2 is not intrinsically simpler than tan2, which is equally correct. This kind of simplicity is relative to a system of notation and highly context-dependent. -- Mike Hardy
No, I agree, in fact tan2 you would probably say is SIMPLER than sec2; more people know what the tan function is than the sec function. My point was not to give a condition that will always produce the simplest integral, because obviously it depends on how you do the actual integration. My point was that the simplest integral does not involve an ARBITRARY constant. It is obvious that sec2, sec2 + 1, sec2 + 2, sec2 + 3 etc are all integrals, but surely you would agree that sec2 is the simplest of them all.
I'm going to rewrite this article at the next opportunity. Kidburla2002
This article should be deleted. It does not make any mathematical sence. If one wishes one can make simple names for an arbritrary function. For example define ed(x) to be cos(x)-457. Now ed(x) has a simpler name than cos(x) and it certainly doesn't have a constant. Is then ed(x) the simplest antiderivative of sin(x)? I don't think the phrase "there is no arbitrary constant" makes any mathematical sence. Isn't 0 a constant? Why 0 is less arbitrary than (say) 1? And the whole point of definite integration is that what the constant does not realy matter. It cancels out! One can use any antiderivative. I fail to see where this "concept" is useful.
I am removing this page since it really doesn't make sence (see previous comment). In case that there is an objection here are its contens:
The simplest integral of a function is an integral with the arbitrary constant of integration set to 0. More specifically, it is an integral of the function with no constant terms in it.
The simplest integral is not necessarily unique. For example, if an integral of a function is sec2(x), then another integral is sec2(x) + 1 because we have just set the arbitrary constant to 1. Now, sec2(x) + 1 is tan2(x), and this is also an integral with no constant terms in it.
Therefore, if more than one simplest integral exists, then you can use any simplest integral of the function where a simplest integral is needed. Such uses include the integrating factor method and definite integration.
- no objections here -- Tarquin 23:47 Jan 21, 2003 (UTC)
No objections here either. Pages that link here need to be fixed though. AxelBoldt 22:12 Mar 8, 2003 (UTC)