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What integration technique?

  1. Oct 18, 2007 #1
    Which integration technique should I use for something similar to:

    (1 + x^2)^(1/2) dx
     
  2. jcsd
  3. Oct 18, 2007 #2

    TMM

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    Trig substitution, I'd say.
     
  4. Oct 18, 2007 #3

    HallsofIvy

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    Staff Emeritus
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    No. All of the "trig substutions" are based on sin^2+ cos^2= 1 and so involve either "1-" or "-1".

    Oh, now that was a silly thing for me to say! sin^2(x)+ cos^2(x)= 1 so, dividing through by cos^2(x), we have
    tan^2(x)+ 1= sec^2(x), just as Sleek suggested. (I am delighted that, while pointing out that I was completely wrong, he referred to an earlier post of mine!)

    Looks to me like a hyperbolic substition should work. Since cosh^2(y)- sinh^2(y)= 1, cosh^2(y)= 1+ sinh^2(y). Let x= sinh(y).
     
    Last edited: Oct 18, 2007
  5. Oct 18, 2007 #4
    Even x=tan(m) would work. One would end up with int([sec(m)]^3) dm. This can be integrated using Int By Parts by differentiating sec(x) and integrating sec^2(x). Also, it can be integrated by method HallsofIvy suggested here: https://www.physicsforums.com/archive/index.php/t-156162.html.

    Regards,
    Sleek.
     
  6. Oct 20, 2007 #5

    Gib Z

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    The only down side is that sec^3 is usually quite a labororus integral to calculate :( Hall's hperbolic suggestion is the quickest.
     
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