Author |
Topic: Help with antiderivative (Read 564 times) |
|
Hooie
Newbie
Gender: 
Posts: 29
|
 |
Help with antiderivative
« on: May 7th, 2004, 4:49pm » |
 Quote  Modify
|
Does arctan (e^x) have an antiderivative?
I tried doing it by parts, but once you do it twice, you end up back where you started.
I tried rewriting it as (arctan x)/x and then using parts, but the same thing happens.
:-\ I'm stuck! Can someone help?
|
|
 IP Logged |
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: Help with antiderivative
« Reply #1 on: May 9th, 2004, 11:31am » |
Quote Modify
|
I find the QuickMath.com's online verison of Mathematica handy for these types of problems. Although I have no idea how it did it, you can click here for your rather ugly integral.
|
|
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: Help with antiderivative
« Reply #2 on: May 9th, 2004, 3:02pm » |
Quote Modify
|
I wonder why it seems to integrate [url=http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=calculus&s2=integrate&s3=basic&expression=(sin(x))^(0.5)
&variables=x&showReply=true&prefInput=standard&prefNames=single&fontSize=20&style=traditional][smiley=smallint.gif](sinx)0.5.dx[/url] but not [url=http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=calculus&s2=integrate&s3=basic&expression=(sin(x))^(1/2)
&variables=x&showReply=true&prefInput=standard&prefNames=single&fontSize=20&style=traditional][smiley=smallint.gif](sinx)1/2.dx[/url]
Well, for (sinx)1/2 it sometimes gives no answer (output = input), and sometimes gives an answer of
'-2E[1/2{(pi/2) - x} | 2]', whatever that is.
Nor do I understand the answer for (sinx)0.5.
|
| « Last Edit: May 10th, 2004, 4:12am by ThudnBlunder » |
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: Help with antiderivative
« Reply #3 on: May 10th, 2004, 5:37pm » |
Quote Modify
|
First, for the original integral. The reason QuickMath was able to integrate it with the closed form
x arctan(ex) + (ix/2)(ln(1+iex) - ln(1-iex)) + (i/2)(Li2(-iex) - Li2(-iex))
is because it has a larger stock of "elementary functions" than you do. Li2(x) is the "dilogarithm": [sum]k=1[supinfty] xk/k2.
Concerning [int](sin x)1/2 dx, I always get a response, though if I don't phrase it right, the thing interprets "s", "i", "n" as constants rather than the name of a function. 
If you use the advanced feature, and select "text" output, you can get clues as to what the functions are. For sin(x)1/2, the function is called "elliptice", which apparently means it is an elliptic integral of the second kind, though the notation I would have expected is -2 E(2, [pi]/4 - x/2). The Elliptic integral function of the second kind is defined by
E(k, x) = [int]0x (1 - k2sin2(t))1/2 dt.
(An exponent of -1/2 is an elliptic integral of the first kind: F(k,x). There is also a more complicated third kind).
The 2F1 function is called "hypergeometric2F1". Here is what Mathworld has to say about it.
The problem with using these results is that unless you are intimately familiar with these functions, these anti-derivatives tell you nothing.
As a side-note, all continuous (and many discontinuous) functions have antiderivatives. However only a few of them have nice closed form expressions. You can define an antiderivative for arctan(ex) by the equation f(x) = [int]0x arctan(et) dt. From a calculational standpoint (easy in calculating the values of f(x) for given x), this might be a better formula than the dilogarithm equation above. We have some fairly stable means of estimating integrals. The calculations might actually prove easier by that means than by calculating the dilogarithm (or perhaps not - the dilogarithm series is fairly simple too). But I would almost guarantee it easier to numerically integrate sin(x)0.5 than to calculate the value from the hypergeometric function.
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
SWF
Uberpuzzler
Posts: 879
|
|
Re: Help with antiderivative
« Reply #4 on: May 11th, 2004, 5:55pm » |
Quote Modify
|
Sometimes it is easier to just expand in a series yourself instead of messing around with those less common functions. In this case, the anti-derivative, F(x), turns out to be an easy to use expression (sums are taken over n= 0 to [infty]):
(for x<=0) F(x)= C + [sum](-1)n exp((2*n+1)x) / (2*n+1)2
(for x>0) F(x)= C + [pi]x/2 + [sum](-1)n exp(-(2*n+1)x) / (2*n+1)2
C is that arbitrary constant math teachers love to dock you points for forgetting. Of course, integral between any given limits a and b is F(b)-F(a), and the value of C does not matter.
Note that integral of atan(exp(x)) from -2 to +2 is [pi].
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print