Integrating a $n$-form over a $n$-dimensional oriented manifold $M$ is possible thanks to the formula
$$
\int_U \omega := \int_{\varphi(U)} A_{\omega,\varphi}(x^1,\ldots,x^n)\mathrm{d}\lambda
$$
where $\varphi\colon U\to\varphi(U)$ is a chart, $A_{\omega,\varphi}$ is the unique function on $\varphi(U)$ such that $\varphi_* \omega = A_{\omega,\varphi} \mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^n$, and $\mathrm{d}\lambda$ is the Lebesgue measure on $\varphi(U)$. A partition of unity argument allows us to integrate $\omega$ on all of $M$ (if $M$ is compact or if $\omega$ has compact support), independently of the considered charts. It is an intrinsic notion.
But there is no natural way to integrate functions over $M$. However, given a reference volume form $\omega_0$, there is a bijection between $\mathcal{C}^{\infty}(M)$ and $\Omega^n(M)$ given by $f\mapsto f\omega_0$. This allows us to integrate functions over $M$ if $M$ is compact or if $f$ has compact support, with the definition
$$
I_{\omega_0}(f) := \int f\omega_0.
$$
This notion of integration of functions on $M$ closely depends on the choice of $\omega_0$! It is not intrinsic.
In the case of a Riemannian manifold, there is a natural choice of volume form: the Riemannian volume form. In local coordinates, it can be expressed as
$$
\mathrm{d}vol_g = \sqrt{\det g(x)}\,\mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^n,
$$
and therefore, in this precise context, integrating a function $f$ means integrating the top-form $f \mathrm{d}vol_g$, which reads in local coordinates $f(x) \sqrt{\det g(x)} \mathrm{d}x^1\wedge\cdots\wedge \mathrm{d}x^n$.
So it does not make sense to integrate "a function + a top-form": what makes sense is to integrate the sum of two top forms. It follows that if $f$ is a function, if $\omega$ is a top-form, and if $\omega_0$ is a reference volume form, the only thing that makes sense is $\int f\omega_0 + \omega$. If the reference volume form is given by a Riemannian metric, it reads, in local coordinates
$$
\int_U f\mathrm{d}vol_g + \omega = \int_{\varphi(U)} \left(f(x)\sqrt{\det g(x)} + A_{\omega,\varphi}(x)\right)\mathrm{d}\lambda.
$$