Applying the Cauchy Integration Formula to $\int_{\left|z\right|=4}{\frac{8\sin(z)}{(z-6)z^2}}dz$ In a section on the Cauchy Integration Formula in my complex analysis text, this problem is an exercise:
Evaluate$$\int_{\left|z\right|=4}{\frac{8\sin(z)}{(z-6)z^2}}\,dz$$
I'm failing to see how I can apply the formula to this problem. I've tried expanding the rational function $\frac{8}{(z-6)z^2}$ and applying partial fractions to little avail. I'm not too  well versed in problems of this form to know what to attempt next. I'm failing to see how I can get this function into a form where my point of evaluation for the integration formula falls inside of the boundary $\left|z\right|=4$.

One can write the integral as 
$$\int_{|z| = 4} \frac{f(z)}{z^2} dz$$
where $f$ is the function
$$f(z) = \frac{8 \sin z}{z - 6}$$
The key point is that $f$ is holomorphic on and within the circle of radius $4$, since its only pole is at $z = 6$. Thus, the Cauchy integral formula implies that
$$\int_{|z| = 4} \frac{f(z)}{z^2} dz = 2 \pi i f'(0)$$
