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Help in a primitive

  1. Feb 18, 2009 #1
    Help in a primitive!!!

    1. The problem statement, all variables and given/known data

    Hello guys! Please, I'm really needing help in a primitive... I don't know, maybe it has a simple solution, but I'm tired and blocked on this... Can you give some lights? Here goes the equation:

    [tex]\int\frac{dx}{x^{2}\sqrt{4-x^{2}}}[/tex]

    2. Relevant equations



    3. The attempt at a solution

    I tried substitution of 4-x^2 and of x^2, but none of them work... I also tried by parts, with u'=1/(x^2) and v=1/sqrt(4-x^2), but it looks like it becomes even heavier... Can you help me?

    Thanks to all and to this great site!
     
  2. jcsd
  3. Feb 18, 2009 #2

    Tom Mattson

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    Re: Help in a primitive!!!

    Trig substitution is the obvious best choice here.
     
  4. Feb 18, 2009 #3
    Re: Help in a primitive!!!

    Yes, of course, you're right! Many Thanks! :) I made x=2*sin(t) and I got:

    [tex]\int\frac{dt}{4sin^{2}\left(t\right)}[/tex]
     
  5. Feb 18, 2009 #4
    Re: Help in a primitive!!!

    Ok, I'm stucked again... I tried:

    [tex]\frac{1}{4}\int\frac{sin^{2}\left(t\right)+cos^{2}\left(t\right)}{sin^{2}\left(t\right)}dt[/tex]

    which gave:

    [tex]\frac{t}{4}+\int\frac{cos^{2}\left(t\right)}{sin^{2}\left(t\right)}dt[/tex]

    Any ideas? I tried partial and substitution but it's a mess...
     
  6. Feb 18, 2009 #5

    Dick

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    Re: Help in a primitive!!!

    Try and differentiate cot(x)=cos(x)/sin(x), ok? What do you get?
     
  7. Feb 19, 2009 #6
    Re: Help in a primitive!!!

    I substituted the fraction above by the cot(t) and then I made the primitive by parts, considering

    u'=1 and thus u=t
    v=cot(t) and thus v'=-2cot(t)/((sin(t))^2)

    Then, I tried to develop the following:

    [tex]\int\frac{cos^{2}\left(t\right)}{sin^{2}\left(t\right)}=t\cot^{2}\left(t\right)+\int\frac{2t\cot\left(t\right)}{sin^{2}\left(t\right)}[/tex]

    What do you think about this? I can try to substitute cot(t) by cos(t)/sin(t), but I'll get a (sin(x))^3 in the denominator... The point is that it seems I'm getting a primitive even more complicated...
     
  8. Feb 19, 2009 #7

    Dick

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    Re: Help in a primitive!!!

    You are making this way too complicated. You wanted to find the integral of dt/sin(t)^2. All I was trying to point out is that the derivative of cot(t) is -1/sin(t)^2. Doesn't that make it easy?
     
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