I was working on this integral:
$$\int_{-\infty}^{+\infty} \frac{x \, dx}{1+x^2}$$
Calculations shows that the limits DNE, and therefore the integral diverge. I used Mathematica and found the same result.
But, the integrand is an odd functions, therefore:
$$\forall c \in \Bbb R : \int_{-c}^{+c} \frac{x \, dx}{1+x^2} = 0 $$
So why don't we just say that: $$\int_{-\infty}^{+\infty} \frac{x \, dx}{1+x^2}=\lim_{c\to\infty} \int_{-c}^{+c} \frac{x \, dx}{1+x^2}=0$$ And the same for any other odd functions?