Solve this complex integral using the Cauchy integral formula

Question

∮(z+4) / ((z^2) +2*z+5)) dz C:|z+1-i| = 2 (center points:(-1,1)) where C is a circle of radius 2?

Answer 1

z^2+2z+5 = (z+1+2i)(z+1-2i)

Thus,

(z+4)/(z^2+2z+5) = (1/2+3/4i)/(z+1+2i) + (1/2-3/4i)/(z+1-2i)

The center of the circle is at (-1, 1)

(-1, 2) is a distance 1 from (-1, 1), while

(-1, -2) is a distance 3 from (-1, 1)

Thus, for (1/2+3/4i)/(z+1+2i), the pole is outside of the disk, while for (1/2-3/4i)/(z+1-2i), the pole is in the interior of the disk.

Thus, while (1/2+3/4i)/(z+1+2i) contributes 0 to the contour integral, as the pole is outside the disk, (1/2-3/4i)/(z+1-2i) contributes (1/2-3/4i)*2πi = πi + 3/2π