# 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 sec^{2}(`x`) is that the "simplest integral" because it has no constant term, even though it is the same as tan^{2}(`x`)+1, which *does* have a constant term? Or is tan^{2}(`x`) the "simplest integral", having no constant term, so that tan^{2}(`x`)+1 would not be the "simplest integral", even though it is the same as sec^{2}(`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 sec^{2} and sec^{2} + 1, the sec^{2} 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 sec^{2} is not intrinsically simpler than tan^{2}, 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 tan^{2} you would probably say is SIMPLER than sec^{2}; 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 sec^{2}, sec^{2} + 1, sec^{2} + 2, sec^{2} + 3 etc are all integrals, but surely you would agree that sec^{2} 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 sec^{2}(*x*), then another integral is sec^{2}(*x*) + 1 because we have just set the arbitrary constant to 1. Now, sec^{2}(*x*) + 1 is tan^{2}(*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)