In Petersen's Riemannian Geometry (2016), it is stated on page 8 that any manifold $M^n$ has an open dense subset $O$ with $TO=O\times\Bbb R^n$. Thus it is orientable and one may define the integral of functions on it using the Riemannian volume form of $O$ (with the inherited metric). Then, since $O$ differs from $M$ by a set of measure zero, one can reasonably define $\int_M$ by $\int_O$. This is especially useful if $M$ is nonorientble and one cannot define a Riemannian volume form on it.
I have never seen integration on nonorientable Riemannian manifolds defined this way. What is the proof of the existence of $O$?
Triangulate the manifold, and remove the (n-1)-squeleton : you have a disjoint union of balls (interior of simplices). By the way, you do not need to remove this balls to define integration : just integrate your function on every simplex (with the use of a Riemanian metric) and sum up over the simplices. Note that Lebesgue measure of the boundary of a simplex is $0$.