Did I get these complex integrals right?(Complex Analysis)

[Raziel2701]

Homework Statement

Determine all possible values of the integral $$\int_{\gamma}\frac{z^2 +1}{z(z-1)}dz$$ where $$\gamma$$ is an oriented circle in C\{0,1}.

Homework Equations

The Attempt at a Solution

I dismantled the integrand by using partial fraction decomposition after doing some long division and so I get the equivalent integral:

$$\int_\gamma dz +\int_\gamma \frac{2}{z-1}dz -\int_\gamma \frac{1}{z}dz$$

So I evaluated that integral over the next four contours.
Gamma 1, circle radius 1/2 centered at 1.
Gamma 2, circle centered at origin, radius 1/2.
Gamma 3, circle that did not encircle neither of the singularities.
Gamma 4, circle centered at origin radius 2.

For each contour respectively I got the following results:
G1:4pi(i)
G2:-2pi(i)
G3:0
G4:2pi(i)

Are these correct?

 

[micromass]

It looks mighty fine to me

 

[tiny-tim]

Hi Raziel2701!

(have a pi: π and a gamma: γ )

For each contour respectively I got the following results:
G1:4pi(i)
G2:-2pi(i)
G3:0
G4:2pi(i)

Are these correct?

Looks good!

except, what if γ is the other way round?

 

[Raziel2701]

If gamma ran counterclockwise, then answers would differ by a negative factor no?

 

[tiny-tim]

(a negative factor? you mean -1 !)

yes