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Smple integral

  1. Mar 28, 2007 #1
    for the life of me I cannot recall how to integrate say

    Int(dv/(k-v^2))=dt


    Its been 30 years since I took calc, and can't find my Taylor book.
     
  2. jcsd
  3. Mar 28, 2007 #2
    Try the substitution v=k1/2sin(x)
     
  4. Mar 29, 2007 #3

    VietDao29

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    Why is there a dt thingy following the integral expression? So, I assume that you want to integrate:
    [tex]\int \frac{dv}{k - v ^ 2}[/tex]
    If k > 0, you can do something like this:
    [tex]\int \frac{dv}{k - v ^ 2} = \int \frac{dv}{(\sqrt{k} - v) (\sqrt{k} + v)} = \frac{1}{2 \sqrt{k}} \int \frac{(\sqrt{k} - v) + (\sqrt{k} + v)}{(\sqrt{k} - v) (\sqrt{k} + v)} dv[/tex]
    [tex]= \frac{1}{2 \sqrt{k}} \int \left( \frac{1}{\sqrt{k} + v} + \frac{1}{\sqrt{k} - v} \right) dv = ...[/tex]
    Can you go from here? :)
    If k < 0 (or in other words, -k > 0), then you can pull out the minus sign, like this:
    [tex]\int \frac{dv}{k - v ^ 2} = - \int \frac{dv}{- k + v ^ 2} = - \int \frac{dv}{(\sqrt{- k}) ^ 2 + v ^ 2} = -\frac{1}{\sqrt{-k}} \arctan \left( \frac{x}{\sqrt{-k}} \right) + C[/tex], can you get this? :)
     
  5. Mar 29, 2007 #4
    sure--er, I think... in the first case, let u=(sqrt(k)+v)
    du=dv then becomes, int(du/u)... =ln(sqrt(k)+v)-ln(sqrt(k)-v)+C=


    using deletes hint, v=k^1/2sin:
    k^-1/2int(cos(x)/(k(1-sin^2(x)) which can be treated similarly or simply looked up as the integral of secant.

    Thanks so much, the memories are coming back.

    Can anyone recommend a good calculus text? I still have my differential eqns/engineering math text, just need a good basic text..
     
    Last edited: Mar 29, 2007
  6. Apr 6, 2007 #5
    Plz, help :

    int_{- infinity}^{+ infinity} dk [exp(ikx)]/(k^2+a^2) -?
     
  7. Apr 6, 2007 #6

    Gib Z

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