Suppose that the coefficientes of an $n-1$ form in $\mathbb R^n$ vanishes in the complement of a compact $K \subset \mathbb R^n$, i.e. if $\alpha = f_1 \,dx_2 \wedge \dots \wedge dx_n \,+\, f_2 \, dx_1 \wedge dx_3 \wedge \dots \wedge dx_n \, + \, f_n \, dx_1 \wedge \dots \wedge dx_{n-1}$, we have that $f_j(x) = 0, \forall x \in \mathbb R^n \setminus K$.
I'm trying to proof that $$\int_{K} d\alpha = 0$$
My attempt was $$ d\alpha = \sum_{j=1}^n (-1)^{j-1} \frac{\partial f_j}{\partial x_j} \, dx_1 \wedge \dots \wedge dx_n \Rightarrow \int_K d\alpha = \sum_{j=1}^n (-1)^{j-1} \int_K \, \frac{\partial f_j}{\partial x_j}\, dm$$
but I can't see why it would be zero.
Help?