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= \frac{-\pi i}{3}\\  \end{align*}  Hence, $\int_{c[-2,3]} \frac{dz}{z^2-2z-8}=\frac{-\pi i}{3}$  By Theorem 4.23(Cauchy's 4.23 (Cauchy's  Theorem): if f is holomorphic in $mathbb\{C}\setminus\left\{-2\right\}$ $mathbb{C}\setminus\left\{-2\right\}$  and