The fundamental theorem of contour integration says if one has a function and its antiderivative, and integrates the function over a closed loop the result is zero.
Cauchy's theorem (Goursat's Version) says the integral of a function in a holomorphic domain in a closed loop is zero.
Cauchy's theorem is apparently much stronger, the proof is certainly more intricate. Can someone please give trivial and nontrivial examples of integrals that Cauchy's Theorem applies to that FTCI does not?
Having proven Cauchy's Theorem, is the FTCI useful for anything, anymore?