Evaluate  ∮_C Re(z) dz

Let C be the unit square with diagonal corners at −1 − i and 1 + i .
Evaluate

∮_C Re(z) dz

Assuming counterclockwise orientation, split the contour into four segments and add up these results for the final answer.

(i) -1-i to 1-i: parameterize via z = x - i with x in [-1, 1].
So, we obtain ∫(x = -1 to 1) (x + i) * 1 dx = 2i.

(ii) 1-i to 1+i: parameterize via z = 1 + iy with y in [-1, 1].
So, we obtain ∫(y = -1 to 1) (1 - iy) * i dy = 2i.

(iii) 1+i to -1+i: parameterize via z = x + i with x in [-1, 1] and opposite orientation.
So, we obtain -∫(x = -1 to 1) (x - i) * 1 dx = 2i.

(iv) -1+i to -1-i: parameterize via z = -1 + iy with y in [-1, 1] and opposite orientation.
So, we obtain -∫(y = -1 to 1) (-1 - iy) * i dy = 2i.

From (i)-(iv), the integral equals 8i.

I hope this helps!