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**manjohn12** **Compute all possible values** of $\displaystyle \int_{C} \frac{z+1}{z(z-1)(z-2)} \ dz $. Assume $\displaystyle C $ is positively oriented and none of the points $\displaystyle 0,1,2 $ lie on the simple closed curve $\displaystyle C $.

So deform $\displaystyle C $ into $\displaystyle C_1, C_2, C_3 $. Then we have $\displaystyle \int_{C} \frac{z+1}{z(z-1)(z-2)} \ dz = \int_{C_1} \frac{z+1}{(z-1)(z-2)} \ dz + \int_{C_2} \frac{z+1}{z(z-2)} \ dz + \int_{C_3} \frac{z+1}{z(z-1)} \ dz $.

This is equaled to: $\displaystyle 2 \pi i(1/2)+ 2 \pi i (-2) + 2 \pi i (3/2) $.

Is this correct?