<s>[INST] Not understanding solution to $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz$ computation Not understanding solution to $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz$ computation.
What was  shown in class:  $\large \int_{|z-2|=2} \frac {5z+7}{z^2+2z-3}dz=\large \int_{|z-2|=2} \frac {5z+7}{(z+3)(z-1)}=\cdots= \int_{|z-2|=2} \frac {2}{z+3}dz+\int_{|z-2|=2} \frac {3}{z-1}dz $
And then it is said that $\large \int_{|z-2|=2} \frac {2}{z+3}dz=0$ while $\large \int_{|z-2|=2} \frac {3}{z-1}dz=3 \cdot 2 \cdot \pi \cdot i$.
Can you please help me understand the last step? What's the difference bewteen the integrals?     
 [/INST] Cauchy's Integral Formula states $$ \oint_C \frac{g(z)}{z-a}dz =g(a)2\pi i.$$ for every $a$ in the interior of $C$. 
Here $g(z) = \dfrac {5z+7}{(z+3)}$, and $a= 1$ hence $$ \oint_C \frac{g(z)}{z-1}dz = \dfrac {5(1)+7}{(1+3)}2\pi i = 6\pi i.$$
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