This is a somewhat simple question: consider a complex manifold $M$ and its canonical bundle $\omega_X$. It is clear that in $H^2(X,\mathbb{R})$, $$c_1(\omega_X) =  c_1(T_X)$$ (Obvious using ChernWeil theory). Does this remain true in $H^2(X,\mathbb{Z})$ ? If not, is there a way to relate the two ?

Yes. This is true for every vector bundle. By functorialuty, it is sufficient to check on just the infinite Grassmanian. But its integral cohomology is torsionfree, so ChernWeil works. 

