  We investigate the finite temperature properties of the one-dimensional
two-component Bose gas (2CBG) with repulsive contact interaction in a harmonic
trap. Making use of a new lattice embedding for the 2CBG and the quantum
transfer matrix we derive a system of two nonlinear integral equations
characterizing the thermodynamics of the uniform system for all values of the
relevant parameters: temperature, strength of the interaction, chemical
potential and magnetic field. This system allows for an easy numerical
implementation in stark contrast with the infinite number of equations obtained
by employing the thermodynamic Bethe ansatz. We use this exact solution coupled
with the local density approximation to compute the density profiles and local
density correlation function of the inhomogeneous gas for a wide range of
coupling strengths and temperatures. Our results show that the polarization in
the center of the trap influences heavily the local correlator especially in
the experimentally accessible Tonks-Girardeau regime.
