<s>[INST] principal value of improper integrals How can I find the principal value of the following? 
$$PV\int_{-\infty}^\infty \frac{e^{ix}}{x(x^2+x+1)} dx$$ 
I'm able to evaluate integrals which have trig identities in them or just polynomials but I can't quite get my head around how to do them with exponentials. Any help would be wonderful please! 
 [/INST] $$\begin{eqnarray*}PV\int_{-\infty}^{+\infty}\frac{e^{ix}}{x(1+x+x^2)}\,dx = PV \int_{0}^{+\infty}\left(\frac{2i(1+x^2)}{1+x^2+x^4}\cdot\frac{\sin x}{x}-\frac{2}{1+x^2+x^4}\cos(x)\right)\end{eqnarray*}$$
and both integrals giving the real and imaginary part can be computed by using the Laplace transform. We have:
$$ \int_{0}^{+\infty}\frac{2\cos x}{1+x^2+x^4}\,dx = \frac{\pi}{3}e^{-\frac{\sqrt{3}}{2}}\left(3\sin\frac{1}{2}+\sqrt{3}\cos\frac{1}{2}\right)$$
and:
$$ \int_{0}^{+\infty}\frac{1+x^2}{1+x^2+x^4}\cdot\frac{\sin x}{x}\,dx = \frac{\pi}{6}\left(3+2\sqrt{3}\,e^{-\frac{\sqrt{3}}{2}}\sin\frac{1}{2}\right).$$
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