I tried my luck with Wolfram Alpha, with $p \in \mathbb{R}$
$$\int_{-\infty}^{\infty} \frac{x^p}{1+x^2} dx = \frac{1}{2} \pi ((-1)^p+1) \sec(\frac{\pi p}{2})$$ for $-1 , and doesn't exist for other $p$. I wonder how to integrate it myself? Especially given that $(-1)^p$ may be a non-real complex number. Thanks in advance! PS: Does Mathematica or some other (free) CAS give the process of deriving the result?
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– 2012-02-21