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Integral of 1/{x-(1-x^2)0.5}

  1. Jun 26, 2011 #1
    1. The problem statement, all variables and given/known data
    [itex]\int[/itex][itex]\frac{dx}{x-\sqrt{1-x^2}}[/itex]


    2. Relevant equations

    3. The attempt at a solution

    i try to simplify it by multiplying [itex]\frac{x+\sqrt{1-x^2}}{x+\sqrt{1-x^2}}[/itex],becoming =[itex]\frac{x+\sqrt{1-x^2}}{2x^2-1}[/itex]
    =[itex]\int[/itex]{[itex]\frac{x}{2x^2-1}[/itex]+[itex]\frac{\sqrt{1-x^2}}{2x^2-1}[/itex]}dx

    ps:
    a.Can i partial fraction the last term to [itex]\frac{A}{x-\frac{1}{\sqrt{2}}}[/itex]+[itex]\frac{B}{x+\frac{1}{\sqrt{2}}}[/itex]??
    b.i try to integrate by using wolfram alpha online ,but the steps is incredibly long..:yuck:
    does it exist any other simpler way??
     

    Attached Files:

  2. jcsd
  3. Jun 26, 2011 #2

    Char. Limit

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    Gold Member

    I checked W-A, and they did it the same way I would have: Begin with a trig substitution like x=sin(u), then use the Weierstrass substitution: v=tan(u/2). It will be long and difficult, but it's possible.
     
  4. Jun 26, 2011 #3

    eumyang

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    Homework Helper

    No, because both the numerator and denominator of the original fraction must be polynomials, and in
    [tex]\frac{\sqrt{1-x^2}}{2x^2-1}[/tex]
    the numerator is not a polynomial.
     
  5. Jun 26, 2011 #4
    o,i see.Thanks a lot:smile:
     
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