Integral of exponential integral
For real nonzero values of x, the
exponential integral
$\;\mbox{Ei}(x)\;$ may be defined as:
$$
\mbox{Ei}(x) = \int_{-\infty}^{x} \frac{e^t}{t}\:dt
$$
I have more than one reason to believe in the following conjecture :
$$
\int_{-\infty}^{+\infty} \mbox{Ei}(-x^2/2)\:dx = -2\sqrt{2\pi}
$$
But could only "prove" this result numerically. Can someone provide a "real" proof?