Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then:

Question

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then:

$$\int_{\gamma_1(I)} \omega = \int_{\gamma_2(I)}\omega.$$

Then, conclude that $H^1(M) = 0$ if $M$ is simply connected.

I saw this claim on a book of topologic degree theory, and I was not able to prove. I do appreciate any help!