  In this paper, we study the higher integrability for the gradient of weak
solutions of $p(x)$-Laplacians equation with drift terms. We prove a version of
generalized Gehring's lemma under some weaker condition on the modulus of
continuity of variable exponent $p(x)$ and present a modified version of
Sobolev-Poincar\'{e} inequality with such an exponent. When $p(x)>2$ we derive
the reverse H\"older inequality with a proper dependence on the drift and force
terms and establish a specific high integrability result. Our condition on the
exponent $p(x)$ is more specific and weaker than the known conditions and our
results extend some results on the $p(x)$-Laplacian equations without drift
terms.
