I'm reading Boas' book and there's a part where she assumes that the following integral goes to $0$ as $\rho$ tends to $\infty$. The argument is that the numerator contains $\rho$ while the denominator contains $\rho^2$. However, I'm uncomfortable with this line of reasoning since the integral has not yet been evaluated.
$\int_0^{\pi} \frac{\rho ie^{i\theta}d\theta}{1+{\rho^2 e^{2i\theta}}}$
Can anybody (a) guide me on how to evaluate the integral (b) tell me why the above reasoning is sound in spite of not evaluating the integral? Perhaps if I could do the first part, it would shed some light on the second. Thanks.