Let $M$ be a compact oriented $k+l+1$ dimensional manifold without boundary in $\mathbb R^n$. Let $\omega$ be a $k$-form and let $\eta$ be an $l$-form, both defined in an open set of $\mathbb R^n$ containing $M$. Show that $$\int_M \omega \wedge d\eta = a \int_M d\omega \wedge \eta$$ for some $a$, and determine $a$.
The professor gave me hint that use the stoke's theorem and the fact that $d(\omega \wedge \eta) = d\omega \wedge \eta + (-1)^k \omega \wedge d\eta$. But I have no idea.
So your Professor's hint is pretty useful!
Do you know what does Stokes' theorem says? If $M$ is a $k+l+1$ manifold then the theorem applies to exact $k+l+1$-forms, such as $d(\omega \wedge \eta)$. That's the left-hand side of the equation your professor gave you.
If you substitute the right-hand side of the equation your professor gave you into Stokes' theorem then you should get an equation of the form you're looking for.