They want me to show this, but I already know that since it is a closed curve, the integral is $0$.
Show that $\int_Cf=0$ where $C:|z|=\sqrt2$ and $f(z)=\frac{(z^2+2)}{z^2+3}$.
Do I just do this stuff? \begin{align*} C&=\sqrt2e^t, t\in(-\pi, \pi)\\ \int_Cf &= \int^\pi_{-\pi} f(C(t))C'(t)dt&u=C(t)\\ &=\int^{-\sqrt2}_{-\sqrt2} f(u)du\\ &=0 \end{align*}
In general, show that the residues of the poles that $C$ encloses sum to zero. In this case, show that $C$ encloses no poles of $f(z)$.