# solve this math question (not homework...HONEST!)

**URL:** <https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999>\
**Category:** Factual Questions\
**Created:** [December 6, 2002, 7:56pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999 "2002-12-06T19:56:51Z")\
**Posts on this page:** 15\
**Page:** 1

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**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\
**Post date:** [December 6, 2002, 7:56pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/1 "2002-12-06T19:56:51Z")

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Take the function X^1/X, and plot it.  
Take the function X^X, and plot that.

What is the area between the two functions, from 0 to 1?

I’m guessing some value related to e, but I never passed calculus.  
(derivatives…first, second, third integrals, inflection points:confused: 😕 😕 )

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [December 6, 2002, 9:01pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/2 "2002-12-06T21:01:54Z")

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The area is most likely infinite, as x[sup]1/x[/sup] goes to infinity as x goes to zero (from the right, which is all that matters here).

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**Author:** ![Taran](https://avatars.discourse-cdn.com/v4/letter/t/edb3f5/32.png) [@Taran](https://boards.straightdope.com/u/Taran)\
**Post date:** [December 6, 2002, 9:16pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/3 "2002-12-06T21:16:26Z")

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hmmm…the problem with this is, X^(1/X) isn’t well-behaved close to zero (as X → 0, 1/X → infinity, so X^(1/X) → infinity). You could measure the area under the curve X^(1/X) from, say, .0000001 to 1, but you can’t get infinitely close to 0 because the closer you get, the bigger the area gets, until you hit 0 and the function stops being well-defined.

That and, I don’t have the foggiest damn idea of how to integrate X^X dx. Or X^(1/X) dx.

Why do you need to know this anyway? 😉

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**Author:** ![Surreal](https://avatars.discourse-cdn.com/v4/letter/s/eada6e/32.png) [@Surreal](https://boards.straightdope.com/u/Surreal)\
**Post date:** [December 6, 2002, 9:18pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/4 "2002-12-06T21:18:16Z")

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I plotted it in Excel, and it looks to me like x^(1/x) goes to zero as x goes to zero.

If it _does_ go to infinity, how would you integrate those functions to find the area from, say, 1/2 to 1.

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**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\
**Post date:** [December 6, 2002, 9:20pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/5 "2002-12-06T21:20:15Z")

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I don’t think so, **ultrafilter**. As a f’rinstance, if x = 1E-n

1E-n[sup]1/1E-n[/sup] = 1E-n[sup]1E+n[/sup]

n = 1, x[sup]1/x[/sup] = 0.1[sup]10[/sup] = 1E-10  
n = 2, x[sup]1/x[/sup] = 0.01[sup]100[/sup] = 1E-200  
n = 3, x[sup]1/x[/sup] = 0.001[sup]1000[/sup] = 1E-3000

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [December 6, 2002, 9:22pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/6 "2002-12-06T21:22:02Z")

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Maybe. It’s still a bitch to integrate, though–you may just have to use numerical means for this one.

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**Author:** ![bryanmcc](https://avatars.discourse-cdn.com/v4/letter/b/eada6e/32.png) [@bryanmcc](https://boards.straightdope.com/u/bryanmcc)\
**Post date:** [December 6, 2002, 9:29pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/7 "2002-12-06T21:29:03Z")

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Incidentally, is there a name for the x^x class of functions? Exponentials? Polynomials? Exponential polynomials?

-b

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**Author:** ![Taran](https://avatars.discourse-cdn.com/v4/letter/t/edb3f5/32.png) [@Taran](https://boards.straightdope.com/u/Taran)\
**Post date:** [December 6, 2002, 9:32pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/8 "2002-12-06T21:32:57Z")

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Doh! (a number \< 1)^(a big honkin’ number) != (another big honkin’ number)

x[sup]1/x[/sup] does indeed approach zero, not infinity. I’m still not sure how to integrate it though. If there is a way, then here’s what you do:

Take the integral from 0 to 1 of x[sup]x[/sup], and subtract the integral from 0 to 1 of x[sup]1/x[/sup]. This works because x[sup]x[/sup] \> x[sup]1/x[/sup] for 0 \< x \< 1 .

I say, just take a bunch of values, put it in Excel, and do it by hand.

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**Author:** ![andy\_fl](https://avatars.discourse-cdn.com/v4/letter/a/49beb7/32.png) [@andy\_fl](https://boards.straightdope.com/u/andy_fl)\
**Post date:** [December 6, 2002, 9:44pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/9 "2002-12-06T21:44:28Z")

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around 0.43

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**Author:** ![Surreal](https://avatars.discourse-cdn.com/v4/letter/s/eada6e/32.png) [@Surreal](https://boards.straightdope.com/u/Surreal)\
**Post date:** [December 6, 2002, 9:47pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/10 "2002-12-06T21:47:33Z")

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I get about 0.435

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**Author:** ![amarinth](https://avatars.discourse-cdn.com/v4/letter/a/3be4f8/32.png) [@amarinth](https://boards.straightdope.com/u/amarinth)\
**Post date:** [December 6, 2002, 10:02pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/11 "2002-12-06T22:02:04Z")

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Doing the - take a bunch of values, throw them in excel, and guess and round, I’m getting ~.43… but that makes a number of wild assumptions, among which is that I remember how to do this at all. (It’s been a while.)

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**Author:** ![Surreal](https://avatars.discourse-cdn.com/v4/letter/s/eada6e/32.png) [@Surreal](https://boards.straightdope.com/u/Surreal)\
**Post date:** [December 6, 2002, 10:05pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/12 "2002-12-06T22:05:15Z")

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You guys did it right.

using 0.001 increments, I get 0.43043

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [December 6, 2002, 10:07pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/13 "2002-12-06T22:07:17Z")

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FWIW, Mathematica (or at least the version on [integrals.com](http://integrals.com)) doesn’t know how to handle those functions. I take that to mean that no one knows.

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**Author:** ![Happy\_Fun\_Ball](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/happy_fun_ball/32/17664_2.png) [@Happy\_Fun\_Ball](https://boards.straightdope.com/u/Happy_Fun_Ball)\
**Post date:** [December 6, 2002, 10:21pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/14 "2002-12-06T22:21:27Z")

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Doing it numerically in Mathematica 4.2 I got 0.429934

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**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\
**Post date:** [December 6, 2002, 10:42pm UTC](https://boards.straightdope.com/t/solve-this-math-question-not-homework-honest/140999/15 "2002-12-06T22:42:27Z")

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My calculator’s usually pretty good about this, and it gets 0.42993371014878. I wouldn’t count on all those decimals being correct, but probably most of them.

It’s still possible that this problem can be solved analytically, even if the indefinite integral of x[sup]x[/sup] cannot. I’m looking for a trick, but no luck yet…