  We generalize, for integral curves, a celebrated result of Max Noether on
global sections of the n-dualizing sheaf of a smooth nonhyperelliptic curve.
This is our main result. We also obtain an embedding of a non-Gorenstein curve
in a way that we can express the dimensions of the components of the ideal in
terms of the main invariants of the curve. Afterwards we focus on gonality,
Clifford index and Koszul cohomology of non-Gorenstein curves by allowing
torsion free sheaves of rank 1 in their definitions. We find an upper bound for
the gonality, which agrees with Brill-Noether's one for a rational and
unibranch curve. We characterize curves of genus 5 with Clifford index 1, and,
finally, we study Green's Conjecture for a certain class of curves, called
nearly Gorenstein.
