Understanding the Cauchy Integral Theorem

My class seems to have only gone so far with the Cauchy Integral Theorem, but maybe I'm just not seeing the other results. For example,

1) One question on the assignment asks if the path integral of C1 and C2 of f(z) are equal, specifically is C1 denotes the positively oriented boundry of the curve given by |x| + |y| = 2 and C2 the positively oriented circle |z| = 4, when f(z) = (z+2) / [sin(z/2)]

2) Evaluating an integral along a square, suggesting that I should be able to choose another curve to integrate on. ex. C = the positively oriented square along the lines x = 2 or -2 and y = 2 or -2, with f(z) = cosh(z) / [z^2 + z]. Evaluate the path integral of f(z) dz

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I'm not asking for answers, per say, but a better understanding of what's going on. These questions seem to reduce to finding and working with singularities, but any general procedure for using CIT would be appreciated, including a statement of a generalized CIT.