The value of the integral $\oint_C(2z+1)dz$ on the contour C What is the value of the integral

Question

The value of the integral $\oint_C(2z+1)dz$ on the contour C, comprised of line segments C1, C2, . . . , C11 shown in Figure

What is the value of the integral? I think we can say (2z+1) is analytic, thus this is an independent from path integral and we can use $\int_{0}^{i}(2z+1)dz$ And our answer will be $i-1$ But how can i say $2z+1$ is analytic. And is my way correct?

Answer 1

Hint: Polynomials are entire. Then what does Cauchy's Theorem tells us?