Help solving complicated integral

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    Integral
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Discussion Overview

The discussion revolves around a complicated integral involving a square root and trigonometric functions. Participants explore methods for solving the integral, share their attempts, and discuss the nature of the integral in relation to elementary and non-elementary functions.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty with the integral and seeks assistance, noting a lack of recent practice in calculus.
  • Another participant requests to see the steps taken so far in attempting to solve the integral.
  • A participant mentions being unable to factor the integral and having tried an online integrator without success, seeking recommendations for software that could provide an answer.
  • One participant suggests that replacing cos² with 1 - sin² leads to an elliptic integral, which cannot be integrated in terms of elementary functions.
  • A later reply indicates that the original poster may not find the integral useful due to unfamiliarity with non-elementary functions.
  • Another participant provides a link to a resource for numerical methods to solve elliptic integrals, cautioning about potential issues with the routines.

Areas of Agreement / Disagreement

Participants do not reach a consensus on how to solve the integral, with some suggesting it leads to non-elementary functions while others express a desire for numerical solutions. The discussion remains unresolved regarding a definitive method for solving the integral.

Contextual Notes

Participants note limitations in their understanding of non-elementary functions and the challenges of integrating the given expression. There is also mention of potential issues with numerical routines.

elkface
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This integral is not overly complicated, but I am having a little trouble with it - I haven't been in calc for a year almost, and it seems I have lost my integration abilities with it.

Here's the integral:
[tex]\int \sqrt{1 + (a \pi x^{a-1})^{2} \cos^{2}({\pi x^{a}})} dx[/tex]

I figure the best place to start is to factor out the inside expression and get rid of the square root, but even doing this I am lost.

Any help?
 
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Can you show your steps so far on your attempt to solve this integral?

Thanks
Matt
 
Well, to be honest, all I can seem to do is stare at it. I can't factor it, clearly, to get it separated into parts so that I could even attempt to do int. by parts or anything of that sort. I have tried Wolfram online integrator, but that doesn't work because it takes the server too long to process.

If anyone has software that could integrate this, or can recommend some (preferably freeware), I'd appreciate it. I don't need the work - I just need the answer.
 
If you replace [itex]cos^2[/itex] by [itex]1- sin^2[/itex] that becomes an elliptic integral which cannot be integrated in terms of elementary functions.
 
Ok, thanks for the tip. In that case, I will probably not be able to put this integral to any good use, as I am not familiar with most non-elementary functions. Thanks anyway.
 
Here is a link to the Numerical Recipe C routine for solving eliptical integrals. Not sure if if will help you. Take caution when using the NR routines they can be problematic. You have been warned.

http://www.nrbook.com/a/bookcpdf/c6-11.pdf

Thanks
Matt
 

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