$$\int_{C}\frac{\sin\pi(z+1)+\cos\pi z}{(z-1)(z-2)}dz\,\,\,\,\,\,\,\,C:|z|=4$$
My try:
Applying Cauchy's Integral formula
$$2\pi i\bigg[\frac{\sin\pi(z+1)+\cos\pi z}{z-1}\bigg]_{z=2}+2\pi i\bigg[\frac{\sin\pi(z+1)+\cos\pi z}{z-2}\bigg]_{z=1}=2\pi i+2\pi i=4\pi i$$
My attempt is correct?