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You should know that using the residue theorem would be easier, but if we are to restrict ourselves to only Cauchy's integral formula, then here's one way of attacking it: First, note that the integrand is a quotient of two entire functions. As such, the integrand is analytic everywhere except the points at which the denominator is zero. Since $z^2+4$ can be factored as $(z-2i)(z+2i)$, then the only points at which the integrand is not analytic are $\pm 2i$.
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