Evaluate where is the circle with positive orientation.

So we have . The last integral is by Cauchy-Goursat. For the two other integrals why can't we say that they are both by Cauchy-Goursat? Because they both have singularities at .

Only the first one is and the second one is by Cauchy Integral Formula. But thet are both analytic within and on . So why are they not both ?

yeah you had to use Cauchy Integral formula for derivatives because the function was "too singular."