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Homework Help: How to approach this integral

  1. Aug 30, 2010 #1

    Mentallic

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    Homework Helper

    1. The problem statement, all variables and given/known data
    [tex]\int\sqrt{1+cos^2x}dx[/tex]



    3. The attempt at a solution
    This problem is part of a bigger picture, and I can't seem to figure out how to approach this integral.
     
  2. jcsd
  3. Aug 30, 2010 #2
    Re: Integral

    It's expressible in terms of the incomplete elliptic integral of the second kind:

    [tex]
    E(\phi, k) = \int_{0}^{\phi}{\sqrt{1 - k^{2} \, \sin^{2} t} \, dt}
    [/tex]

    Hint:

    Express the cosine squared in terms of sine squared and then divide by the free term under the squared root to
     
  4. Aug 30, 2010 #3

    Mentallic

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    Re: Integral

    In other words, not expressible in terms of elementary functions. Looks like my calculator is of no use then, gay...

    So the answer is [tex]\sqrt{2}E\left(1,\frac{1}{2}\right)[/tex]

    How could I go about finding an approximation for this?
     
  5. Aug 30, 2010 #4
    Re: Integral

    Here's the approximation according to Mathematica to 50 decimal places:

    1.3114424982155470455454946537619651179489905076619

    Code (Text):
    N[Sqrt[2] EllipticE[1, 1/2], 50]
    is the command used...
     
  6. Aug 30, 2010 #5
    Re: Integral

    I think you made a mistake. FIrst of all, your [itex]k[/itex] is wrong. Secondly, I don't know how you found that upper limit, since you had an indefinite integral.
     
  7. Aug 30, 2010 #6

    Mentallic

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    Re: Integral

    Nope, I'm fairly certain my k is correct and I originally posted the indefinite integral assuming I wouldn't need help with evaluating the limits, they were 0 to 1 as you'd expect.
     
  8. Aug 30, 2010 #7
    Re: Integral

    Ok then, cool. Have a nice life.
     
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