Let $\theta,\gamma$ be $3-$forms on $\Bbb{S}^7$. show that
$$\int_{\Bbb{S}^7} \theta \, \wedge d \gamma = \int_{\Bbb{S}^7} d \theta \, \wedge \gamma$$
My thoughts: Subtract the LHS from the RHS to show
$$\int_{\Bbb{S}^7} d \theta \, \wedge \gamma + (-1) \theta \, \wedge d \gamma= \int_{\Bbb{S}^7} d(\theta \wedge \gamma)=\int_{\partial \Bbb{S}^7} \theta \wedge \gamma = 0$$
Where the first equality is by the anti derivation property of the exterior derivative, the second equality is by Stoke's Theorem, the last equality is because the boundary of $\Bbb{S}^7$ is the empty set thus the integral is zero. Is this correct? And we have that $\theta \wedge \gamma$ is a $7-1=6-$form on $\Bbb{S}^7$ thus Stokes Theorem can be applied.