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Possible Duplicate:
Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?

How to prove $\operatorname{si}(0) = -\pi/2$ without contour integration ? Where $\operatorname{si}(x)$ is the sine integral.

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    I think you mean $Si(\infty)$ or $-si(0)$. See http://en.wikipedia.org/wiki/Sine_integral#Sine_integral2012-09-05

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HINT:

Note that our integral may be rewritten as $\int_{0}^{\infty} \int_{0}^{\infty} e^{-xy} \sin x \ dy \ dx = \int_{0}^{\infty} \frac{\sin x}{x} \ dx$ but integrating with respect to x we get that $\int_{0}^{\infty} \int_{0}^{\infty} e^{-xy} \sin x \ dx \ dy = \int_{0}^{\infty} \frac{1}{1+y^2} \ dy$ Hence I hope you can handle it on your own.