Question 8

Let $${\alpha = \frac{z dz}{(z^2 + 1)(z - 4)^2}}$$.
Let $\Gamma$ be a circular contour in the plane, avoiding all singularities of $\alpha$.

• Find all possible results of the integral of $\alpha$ around the contour $\Gamma$, taken once counter-clockwise.
• Is there a region of area $10^{20}$ in which the differential $\alpha$ is exact?
  Explain your answer.