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This article concerns the higher integrability of a very weak solution $u\in \theta+ W_0 ^{1,r}(\Omega)$ for $\max \{1,p-1\}<r<p<n$ to the Dirichlet problem of the nonlinear elliptic system $$\displaylines{ -D_\alpha\mathbf{A}_i^\alpha(x,Du)= \mathbf{B}_i(x,Du) \quad \text{in }\Omega,\cr u=\theta \quad \text{on } \partial\Omega, }$$ where $\mathbf{A}(x,Du)=\big(\mathbf{A}_i^\alpha(x,Du)\big) $ for $\alpha=1,\dots,n$ and $i=1,\dots,m$, and each entry of $\mathbf{B}(x,Du)=\big(\mathbf{B}_i(x,Du)\big)$ for $i=1,\dots,m$ satisfies the monotonicity and controllable growth. If $\theta \in W^{1,q}(\Omega)$ for q>r, then we derive that the very weak solution u of above-mentioned problem is integrable with $$ u\in \cases{ \theta +L_{\rm weak}^{q^*} (\Omega) & for $1\le q<n$,\cr \theta +L^\tau(\Omega) & for $q=n$ and $1<\tau<\infty$,\cr \theta +L^\infty (\Omega) & for $q>n$, } $$ provided that r is sufficiently close to p, where $q^*=qn/(n-q)$.