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 October 14th, 2012, 10:18 PM #1 Senior Member   Joined: Jul 2011 Posts: 395 Thanks: 15 Indefinite Integration $\displaystyle \int\frac{1}{\sqrt[4]{1+x^4}}dx$
 October 14th, 2012, 11:08 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 461 Math Focus: Calculus/ODEs Re: Indefinite Integration This is not expressible in elementary terms.
October 15th, 2012, 02:26 AM   #3
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Re: Indefinite Integration

Quote:
 Originally Posted by MarkFL This is not expressible in elementary terms.
I don't agree !
Let x = ((t^4)-1)^( -1/4)
Bring it back into the integal. This leads to an integral easy to express in terms of usual functions.

October 15th, 2012, 04:37 AM   #4
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Re: Indefinite Integration

Thanks JJacquelin and Markfl

My solution::

[attachment=0:1705p6tf]Integral......gif[/attachment:1705p6tf]
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 Integral......gif (10.1 KB, 202 views)

 October 15th, 2012, 05:14 AM #5 Senior Member   Joined: Aug 2011 Posts: 333 Thanks: 8 Re: Indefinite Integration Your transformation is not completed yet because there is still (x^4) remainings at denominator. No x must remain in the integral. This is an integral with only t.
October 15th, 2012, 11:32 AM   #6
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Re: Indefinite Integration

Quote:
 Originally Posted by JJacquelin Your transformation is not completed yet because there is still (x^4) remainings at denominator. No x must remain in the integral. This is an integral with only t.
Try scrolling down, that's what I thought at first.

@panky
That is the substitution that came into my mind too!

Regards,
Rejjy
16-Oct-2012
01:02 IST

 October 15th, 2012, 11:41 AM #7 Math Team     Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions Re: Indefinite Integration Hi guys .. This substitution is giving me a headache .. There is something wrong with it ! I agree With Mark
October 15th, 2012, 12:17 PM   #8
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Re: Indefinite Integration

Quote:
 Originally Posted by zaidalyafey Hi guys .. This substitution is giving me a headache .. There is something wrong with it ! I agree With Mark
Nothing is wrong. You just have to continue the work done by panky (8:36 am)
It remains only to find the primitives of -t²/((t^4)-1), which is not too difficult.

October 16th, 2012, 01:36 AM   #9
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Re: Indefinite Integration

Quote:
 Originally Posted by zaidalyafey Hi guys .. This substitution is giving me a headache .. There is something wrong with it ! I agree With Mark
panky has already solved it, why are you confused? The integral is expressed as elementary form.
Here are my initial steps which prompted me of the substitution.
$I=\frac{1}{(1+x^{\small{4}})^{1/4}}
I=\frac{1}{x(x^{\small{-4}}+1)^{1/4}}
I=\frac{x^4}{x^5(x^{\small{-4}}+1)^{1/4}}
\text{Now substitute }
x^{\small{-4}}+1=t^4
1+x^4=t^4x^4$

Regards,
Rejjy
16-Oct-2012
15:02 IST

 October 16th, 2012, 02:14 AM #10 Math Team     Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions Re: Indefinite Integration You are right guys ,,, I agree with you

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