We consider the Cauchy problem of the KdV-type equationPilod (2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space H s (R) for any s ∈ R if c 1 = 0. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in H 2 (R) with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in H 1 (R) with bounded primitives.