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∫sqrt (1+sec^4(x)) dx

  1. Jun 9, 2007 #1
    1. The problem statement, all variables and given/known data

    solve : ∫ sqrt (1+sec^4(x)) dx


    2. Relevant equations



    3. The attempt at a solution

    I tried to do the substitution rule but it makes the problem more complicated
     
  2. jcsd
  3. Jun 9, 2007 #2

    Office_Shredder

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    What substitution did you use?
     
  4. Jun 9, 2007 #3
    U= sec(x)
    dU= sec (x) tan (x) dx

    I think it's wrong anyway
     
  5. Jun 9, 2007 #4

    mjsd

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    This is not an easy integral, you sure you have got it written down right?
    in fact integrating sqrt(1+x^4) dx or 1/sqrt(1-sin x ^2) is a hard enough problem. for something that looks like sqrt(1+(sec x)^4) you would expect that it is related elliptic integrals.
     
  6. Jun 9, 2007 #5
    mjsd yes I'm sure of it...

    I tried also Simpson method and got = 1.278 but that is just an approximation.
     
  7. Jun 9, 2007 #6

    Gib Z

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    I don't seem to follow you skeem, this is an indefinite integral you proposed to us, yet you are using some sort of bounds for your simpsons rule approximation. Perhaps telling us the bounds could help, as it does in many cases.

    I think the best method would be a substitution t=tan(x/2), however if possible its definitely not easy. The integrator doesn't rule out the possibility, just states that it could not do it in the alloted time..perhaps someone with mathematicia on their computer can do it for us?
     
  8. Jun 9, 2007 #7

    Office_Shredder

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    Maple gives a closed form solution

    [tex]-\sqrt {{\frac {1+\cos \left( x \right) }{\cos \left( x \right) }}}

    \sqrt {-2\,{\frac {\cos \left( x \right) }{1+\cos \left( x \right) }}}

    \sqrt {2}{\it arctanh} \left( 1/2\,\sqrt {2}\sin \left( x \right)

    \sqrt {-2\,{\frac {\cos \left( x \right) }{1+\cos \left( x \right) }}}

    \left( \cos \left( x \right) \right) ^{-1} \right)

    [/tex]

    Thankfully, it lets you convert automatically to latex :)

    It should be noted that when I take the integral of that expression in maple, it just doesn't do it, instead giving me the "symbolic" answer of an integral sign followed up by the expression. Simplify doesn't seem to simplify it very much either
     
    Last edited: Jun 9, 2007
  9. Jun 10, 2007 #8
    Gib Z the bounds are 0 and pi/4

    Office_Shredder :surprised that's horrible solution!! but I really appreciate your help
     
  10. Jun 10, 2007 #9

    Office_Shredder

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    I apologize.... I went back to have maple numerically solve from 0 to pi/4, and realized earlier I had mistyped, and only evaluated the integral of
    [tex][1+sec(x)]^{1/2}[/tex] Maple doesn't actually do the integral you're told in a closed form solution, but the numerical approximation is 1.277978059
     
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