Applying Cauchy-Goursat theorem/ doing it the "hard way"

I must apply Cauchy-Goursat theorem to show that where and .

So what I wrote was "f is analytic over " therefore according to C-G's theorem the line integral is worth 0 since the singularity of f isn't inside the enclosed (by C) domain. Yet I don't feel I've "showed" anything. Do you think it's enough?

So I even tried to directly show that . I parametrized the unit circle as for . I came up with that this integral is worth . Then I've used Euler's formula and couldn't show anything. (I get a very long and ugly impossible thing). How would you show that the integral is worth 0 using only the definition of a line integral?