# 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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