$f(z) =\frac1{z^2-6z+8}$ then Evaluate the following integral: $\oint\limits_{C} u dy +v dx$ where $C$ is $|z|=3$

Question

Let $u$ and $v$ be the real and imaginary parts respectively of the function $f(z) =\frac1{z^2-6z+8}$ of a complex variable $z=x+iy$. Let $C$ be the simple closed curve $|z|=3$ oriented in the counter clockwise direction. Evaluate the following integral: $\oint\limits_{C} u dy +v dx$
$\bullet$ I'm confused whether I can use residue theorem or not? Please help. Thanks in advance.

Answer 1

Taking $f(z)=u+iv, dz=dx+idy,$ note that $$udy+vdx=\Im \oint f(z) dz=-\pi$$ The integral has been evaluate by Residue theorem for the pole at $z=2$ as $|z|<3$