I'm trying to show that if $\rho = 0$ on the bivariate normal, X and Y are independent. My approach is to show $f(x, y)=f_{X}(x) f_{Y}(y)$
Then $f_{X}(x) = \int_{-\infty}^{\infty}{f(x, y) \ dy}$ and same for $f_{Y}(y)$