Dominated convergence theorem for singular function

Question

Suppose I have a family of integrable function $f_a$ such that $\int_\Omega f_a <M$ and this $M$ is independent of $a$. And $f_a$ is smooth in $\Omega$ but $f_a|_{\partial \Omega}=\infty$ for all $a$. Also , $f_a$ ia locally uniform bounded and $f_a$ converges to $f_b$ pointwisely in any compact subset of $\Omega$. Could I conclude that $$\lim\limits_{a \rightarrow b}\int_\Omega f_a=\int_\Omega f_b$$