List of top Complex Analysis Questions on Contour Integration and Cauchy's Theorem

The value of the contour integral \[ \int_{\Gamma} \frac{(1-z)(3\cos z+5\sin z)}{1-z^{101}}\,dz, \] where \(\Gamma = \left\{z\in\mathbb{C} : |z|=\frac{7}{9}\right\}\), oriented in the counter-clockwise direction, is equal to