  The weight of a coset of a code is the smallest Hamming weight of any vector
in the coset. For a linear code of length $n$, we call integral weight spectrum
the overall numbers of weight $w$ vectors, $0\le w\le n$, in all the cosets of
a fixed weight. For maximum distance separable (MDS) codes, we obtained new
convenient formulas of integral weight spectra of cosets of weight 1 and 2.
Also, we give the spectra for the weight 3 cosets of MDS codes with minimum
distance $5$ and covering radius $3$.
