Suppose I have integrable function sequences $f_n(x)$ and $g_n(x)$ and we know that point wisely, $f_n(x)\to g_n(x)$, to be more specific, $\forall x$ and $\forall \epsilon>0$, $\exists N$ such that $\forall n>N$, we have
$$|f_n(x)-g_n(x)|<\epsilon.$$
and that both $f_n(x)$ and $g_n(x)$ are dominated by some integrable function $h(x)$. Can we conclude via some version of DCT that
$$\int f_n(x)dx \to \int g_n(x)dx?$$
For each $n$, $$\left\lvert\int f_n-\int g_n\right\rvert\le\int\lvert f_n-g_n\lvert$$
As you've written, $\lvert f_n-g_n\rvert\to 0$ pointwise and it is dominated by $\lvert h_f\rvert+\lvert h_g\rvert$. Therefore the RHS converges to zero and $$\lim_{n\to\infty} \left(\int f_n-\int g_n\right)=0$$
regardless of even either (and thus both) of the integrals actually converging.