Connected, compact, orientable smooth manifold with finite fundamental group The problem is prove that a connected, compact, orientable smooth manifold $M$ with finite fundamental group has $H^{1}_{dR}(M)=0$. The hint is to apply

Suppose $\tilde M$ and $M$ are smooth $n$-manifolds and $\pi:\tilde M\to M$ 
  is a smooth $k$-sheeted covering map. If $\tilde M$ and $M$ are oriented and $\pi$ is orientation-preserving, show that $\int_{\tilde M}\pi^*\omega=k\int_{M}\omega$ for any compactly supported $n$-form $\omega$ on $M$.

but i don't know how this proves the result! 
Any tips?

I honestly don't know what I was thinking when I wrote that hint -- it doesn't seem to be useful at all. 
A more appropriate hint would have been something like this: "If $\omega$ is a closed $1$-form on $M$, let $\widetilde\omega$ be the pullback of $\omega$ to the universal cover of $M$, and let $\tilde f$ be a potential function for $\omega$. Define $f\colon M\to \mathbb R$ by letting $f(x)$ be the average value of $\tilde f$ over the preimages of $x$, and show that $f$ is a smooth potential function for $\omega$."
This answer by @TedShifrin gives a little bit more detail about how to carry out that argument. 
There's a different proof of this result in the second edition of my Introduction to Smooth Manifolds -- see Corollary 17.18.
