This is a relation which seems to make sense but I couldn't find anything on google.
$$\int^a_b \int^c_d \alpha f(x)g(y)~dxdy= \alpha \int^a_b g(y)dy\int^c_d f(x)dx$$
It just seems less likely to make mistakes when splitting the integrals like this as you don't have to even think about $y$ when integrating w.r.t $x$. The only caveat I can think of is if instead of constant limits of integration we have maybe a function of $x$ or $y$ in the limits of integration then presumably we have to follow whatever order of integration is needed to compute the integral?
Thanks.