Integral of a k-form over an oriented compact manifold

Question

I see in my course the following theorem:

If $\omega$ is an exact k-form over an oriented compact manifold M of dimension $k$, then $\int_M \omega=0$.

I don't have a proof of this theorem and I only know it's an application of stokes theorem.

Is this theorem correct?

Thank you for any help.

Answer 1

$$ \text{With }\omega = d\eta, \qquad \int_M \omega = \int_M d\eta = \int_{\partial M} \eta = 0 \quad \text{since }M \text{ has no boundary.}$$