$$pv\int_{-\infty}^{\infty} \frac{x}{(x^2+4)(x^2+2x+2)}dx$$
I get an answer of $-\frac{\pi}{5}$ but wolframalpha disagrees by a factor of $2$ ($-\frac{\pi}{10}$):
http://www.wolframalpha.com/input/?i=integration+%28z%2F%28%28z^2%2B4%29%28z^2%2B2z%2B2%29%29%29++from+-+infinity+to+infinity
I cannot spot the error.
I get these residues:
$$Res(-1+i)=\frac{1+3i}{20}$$
$$Res(2i)=\frac{-2-i}{40}$$
which I add up and multiply by $2\pi i$.
What am I doing wrong?
Answer
$$\text{Res}(2i)={x\over (x+2i)(x^2+2x+2)}\Big{|}_{x=2i}={2i\over4i(4i-2)}={1\over4(-1+2i)}={-1-2i\over20}$$
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