I want to prove the following claim :
If $(M,g)$ is a compact Riemannian manifold , then for any smooth $f:M\to\mathbb{R}$ , one has $$\int_M\Delta f\,*(1)=\int_M\Delta f\sqrt{|g|}\,dx^1\wedge dx^2\wedge\dots\wedge dx^k=0$$ where $*:\Omega^0(M)\to\Omega^k(M)$ with $k\geq0$ is the linear star operator between space of differential $k$-forms .
I understand that using Gauss' theorem $$(\nabla f,X)=-(f,\nabla.X)+\int_{\partial M}f\langle X,N\rangle\,dS$$ for a compact smooth manifold $M$ with smooth vector field $X:M\to TM$ , and $(\cdot,\cdot)$ being $L^2$ inner product of functions and $\langle\cdot,\cdot\rangle$ being inner product of vectors , $N$ the usual outward unit normal to $\partial M$ . But I am unable to prove the rest . Any help is appreciated .