1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook