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Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) to $X$. Since the integrals for orientable manifolds would coincide with this construction, why don't we just define the integral for non-orientable compact manifolds in this way? I.e. lift a function $X \rightarrow \mathbb{R}$ to a $\pi$-invariant function on the universal cover and integrate it on a fundamental domain, where $\pi=\pi_1(X)$

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    $\begingroup$ Yes We Can. Even easier you can go to the 2-fold orientable cover, integrate your function there, and then divide the result by 2. (This doesn't require to construct a fundamental domain.) $\endgroup$
    – ThiKu
    May 22, 2016 at 20:38

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