I found this identity in an old book on integration, given without any explanation: $$ \int_{-\infty}^{\infty} ze^{-z^2/2}dz\int_{-\infty}^{z} e^{-x^2/2}dx = \int_{-\infty}^{\infty}e^{-x^2/2}dx \int_{-x}^{\infty}z e^{-z^2/2}dz $$ It may be trivial, so the question might be much easier than I imagined, but I really cannot see any jutification for this step.
Any help is greatly appreciated.
edit Apparently, the problem was a typo in the last integral, that preventd me to see how the integration was to be carried over. The last integral should be
$$ \int_{x}^{\infty}z e^{-z^2/2}dz $$ and then the equality is easily established via the Fubini theorem.
The region over which we are integrating is the half of the first quadrant bounded by the lines $x=0$ and $y=x$. Therefore the integrand is a nonnegative function, so Tonelli's theorem applies and the iterated integrals are equal. But it is a good exercise to actually sketch this region, as this often is an effective method of determining the new limits of integration when interchanging the order.