​$$\text{Let } \gamma \text{ be the positively oriented circle in the complex plane given by:} \\\{z \in \mathbb{C} : |z - 1| = 1\}. \\\text{Then:} \\\frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z^3 - 1} \quad \text{equals.}$$​​

Correct option is B

​$$\text{The equation :} \ z^3-1\ \text{, have 3 roots :}\ 1,w,w^2\\So,\\\frac{1}{z^3-1}=\frac{1}{(z-1)(z^2+z+1)}=\frac{1}{(z-1)(z-w)(z-w^2)} \\ .\\ \textbf{The diagram of given contour is :}$$

Among the three poles($$1,w,w^2$$​) only 1 is inside the contour , 

$$\textbf{Using caucy integral formula :}\\\frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z^3 - 1} = \frac{1}{2\pi i} \int_{\gamma} \frac{dz}{(z-1)(z^2 + z + 1)} \\= \frac{1}{2\pi i} \times 2\pi i \left( \frac{1}{z^2 + z + 1} \right) \bigg|_{z = 1} \\= \frac{1}{z^2 + z + 1} \bigg|_{z = 1} = \frac{1}{3}$$​