In one of our classes the teaching assistant argued that $\int_\mathbb{R}ye^{-\frac{y^2}{2}}$ is an odd function and hence the integral is $0$.
But this argument doesn't hold in the case of $\int x \frac{1}{1+x^2}$.
In general when dealing with indefinite integrals I'm not allowed to use the argument odd-function, right?. I have to compute it by hand. So for the first example:
$\int_\mathbb{R}ye^{-\frac{y^2}{2}} = [-e^{-\frac{y^2}{2}}]|_{-\infty}^{+\infty} = 0 - 0 = 0$ would have been the right argument to say that the integral is $0$, right?
If you can determine in advance that the integral over $\mathbb{R}$ is convergent, then you can say without further computation that it must be zero (under the assumption that the integrand is odd, of course).