While estimating the integral, $$\int_{y=0}^{\infty} \int_{x=0}^{\infty} \frac{xy \ln x}{(x^2+1)(x^2 y^2-a^2)(y^2+1)} dx dy \,(a>0) $$, I made a change in the order of integration of the integral. But I am unable to justify why we can change the order of integration. I know Fubini and Tonneli's theorem. But I can't apply them directly. Could someone please tell me whether this change is indeed justified? Any help would be appreciated. Thanks in advance.
I start with $$\int_{y=0}^{\infty} \int_{x=0}^{R} \frac{xy \ln x}{(x^2+1)(x^2 y^2-a^2)(y^2+1)} dx dy , $$ and try to use dominated convergence theorem to interchange limit and integral and Fubini's theorem. I tried checking whether the function $\frac{1}{x^2 y^2-a^2}$ is bounded near $x=0 $ and $x= \infty$. But I can't show the function which bounds the integrand is in $L^{1}$.
(i) For $0<c<d$ try integrating over $ c \le x,y \le d$ and then taking the limit as $c \to 0, d \to \infty.$ (ii)Alternatively try, in polar coordinates, for $0<R_1<R_2,0<\phi<\psi <\pi/2$ integrating over the region $$R_1 \le r \le R_2,\phi \le \theta \le \psi $$ and then taking the limit as $$R_1 \to 0,R_2 \to \infty,\phi \to 0,\psi \to \pi/2.$$