text: # Evaluate this complex Integral
  
  1. May 6, 2012
  
  ### bugatti79
  
  1. The problem statement, all variables and given/known data
  Use Cauchy's Integral Theorem to evaluate the following integral
  
  $\int_0^{\infty} \frac{x^2+1}{(x^2+9)^2} dx$
  
  2. Relevant equations
  
  Res $f(z)_{z=z_0} = Res_{z=z_0} \frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}$
  
  3. The attempt at a solution
  
  I determine the roots of the denominator to be $x=\pm 3i$.
  How do I convert these into polar form. I know $z=re^{i\theta}$
  
  Do I need to convert these into $z=e^{f(i\theta)}$?
  
  2. May 6, 2012
  
  ### Robert1986
  
  I wouldn't convert this to polar form. You need to figure out how to write $f$ as a rational function like $p/q$ that you have above. Where $p$ is analytic and non-zero at $z_0$.
  
  3. May 7, 2012
  
  ### bugatti79
  
  I realise the pole is second order therefore using the residue forumla for second order pole $z=3i$ we get
  
  $=lim_{z \to 3i} \frac{d}{dz} \frac{ z^2+1}{(z^2+9)^2} = \frac{-2z(z^2-7)}{(z^2+9)^3} |_{z=3i}$
  
  but we are going to get 0 in the denominator here..?
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