Say I have 2 continuous bivariate distributions (for example bivariate log normal and bivariate weibull) which have joint density functions as: $f_1(x,y,\theta)$ and $f_2(x,y,\alpha).$
Now I want to do something like:
\begin{align*} \bigg[\int_{0}^{\infty} \int_{0}^{\infty}\frac{\partial}{\partial \alpha}\log\bigg({f_{2}(x,y;\alpha)}\bigg)f_{1}(x,y;\theta) \,dx\,dy\bigg] =\frac{\partial}{\partial \alpha} E_{f_1}(\log f_2(X,Y,\alpha)) \end{align*}
I know by DCT I can do that but to check the condition of the integrand in LHS being dominated by some integrable function of X and Y is a tedious task. Can we check this condition by some alternate way which makes it easier to do so?