I want to obtain I as a function of V, in the following equation
$$I(V) = \int _0 ^\infty\int _0 ^\infty \frac{1}{(1 + e^{x+y})(e^{V-x-y}+1)}\frac{1}{\sqrt{y}}dx dy$$
I would prefer to do this on Mathematica if it is easy. I want to numerically evaluate the integral so as to get a curve of I v/s V.
Context - Basically, the integrand is a product of fermi-functions which i am trying to evaluate so as to get current versus voltage relation in a device.
As Yuriy mentioned, one may integrate over $x$ fairly easily to get that the double integral is equal to
$$I(V)=\frac{2}{e^V-1} \int_0^{\infty} dy \, \log{\left (\frac{1+e^Ve^{-y^2}}{1+e^{-y^2}} \right )} $$
This is a nice, smooth function that decreases very rapidly and which Mathematica should find simple.
Alternatively, depending on the value of $V$, you can Taylor expand the integrand and express the integral as a sum.