A proof of the Cauchy-Goursat theorem
A proof of the Cauchy-Goursat theorem can be found, for instance in the fourth chapter of the book:
Churchill, Ruel V.; Brown, James W.
Complex variables and applications. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg
Click HT25-Cauchy-Goursat.pdf link to view the file.