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Integral help

  1. Oct 18, 2007 #1
    Need help finding

    [tex]\int[/tex]1/ (x[tex]^{}2^{}[/tex] + b[tex]^{}2^{}[/tex] )[tex]^{}n^{}[/tex] dx

    with limits negative infinity to infinity

    where b, n some constant

    No work is required since its just part of a quantum mechanics problem, i cant find the integral in any tables and i dont have mathematica or anything available to me right now.

    edit: or if anyone can link me to a page with definite integrals of that form, as i also may need the same thing multiplied by x^n

    Last edited: Oct 18, 2007
  2. jcsd
  3. Oct 18, 2007 #2
    Unless I'm mistaken, the problem will depend on whether or not n is even or odd.

    let [tex] x= b \tan{\theta} [/tex]

    then [tex] (x^2 + b^2)^n = b^{2n}(\tan^2{\theta} + 1)^n[/tex]
    [tex] dx = b \sec^2 {\theta} [/tex]


    [tex] \displaystyle \int \frac{1}{(x^2+b^2)^n} dx = \frac{1}{b^{2n-1}}\int \frac{d\theta}{\sec^{n-2}{\theta}} d\theta[/tex]

    [tex] = \frac{1}{b^{2n-1}} \int cos^{n-2}{\theta} d\theta[/tex]

    Now for simplicity sake, let [tex] k= n-2[/tex]

    If k is even (iff n is even) then use the identity

    [tex] \cos^2 {x}= \frac{1}{2} ( 1 + \cos{2x}) [/tex]

    If k is odd (iff n is odd) then take

    [tex] \cos^k{x} = \cos{x}(1-\sin^2{x})^{k-1}[/tex] and use basic substitution.

    Note that (k-1) is even since k is odd.
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