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Let $\Phi:TM\to TM$ be a positive-semidefinite symmetric operator of class$C^1$ defined on a complete non-compact manifold $M$ isometrically immersed ina Hadamard space $\bar{M}$. In this paper, we given conditions on the operator$\Phi$ and on the second fundamental form to guarantee that either $\Phi\equiv0$ or the integral $\int_M \mathrm{tr}\,\Phi dM$ is infinite. We will givensome applications. The first one says that if $M$ admits an integrabledistribution whose integrals are minimal submanifolds in $\bar{M}$ then thevolume of $M$ must be infinite. Another application states that if thesectional curvature of $\bar{M}$ satisfies $\bar{K}\leq -c^2$, for some $c\geq0$, and $\lambda:M^m\to [0,\infty)$ is a nonnegative $C^1$ function such thatgradient vector of $\lambda$ and the mean curvature vector $H$ of the immersionsatisfy $|H+p\nabla \lambda|\leq (m-1)c \lambda$, for some $p\geq 1$, theneither $\lambda\equiv 0$ or the integral $\int_M \lambda^s dM$ is infinite, forall $1\leq s\leq p$.

Integral estimates for the trace of symmetric operators