We have studied the anisotropy of the London penetration depth of carbon doped $MgB_2$ single crystals, which was obtained from reversible magnetization measurements with the magnetic field both parallel and perpendicular to the c-axis. Similar to the pure $MgB_2$, the anisotropy of the upper critical field ${gamma}_H$ decrease with temperature while the anisotropy of the London penetration depth ${gamma}_{lambda}$ slowly increases with temperature. However, the temperature dependence of ${gamma}_H$ is drastically reduced and the value of ${gamma}_{lambda}$ becomes nearly ~1 as C is introduced. These results indicate that C substitution increases impurity scattering mainly in the $sigma$ bands. The temperature dependence of the anisotropies agree well with the theoretical predictions with impurity scattering.