Precision measurement of the integrated luminosity of the data taken by BESIII at center-of-mass energies between 3.810 GeV and 4.600 GeV**Supported by National Key Basic Research Program of China (2015CB856700), National Natural Science Foundation of China (NSFC) (11125525, 11235011, 11322544, 11335008, 11425524), Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program, Joint Large-Scale Scientific Facility Funds of the NSFC and CAS (11179007, U1232201, U1332201) CAS (KJCX2-YW-N29, KJCX2-YW-N45), 100 Talents Program of CAS, INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology, German Research Foundation DFG (Collaborative Research Center CRC-1044), Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey (DPT2006K-120470), Russian Foundation for Basic Research (14-07-91152), U.S. Department of Energy (DE-FG02-04ER41291, DE-FG02-05ER41374, DE-FG02-94ER40823, DESC0010118), U.S. National Science Foundation, University of Groningen (RuG) and the Helmholtz