Convergence of $\int_{-\infty}^{\infty}f(x)dx$ I posed a question to my calculus professor, asking how to evaluate the Riemann integral
$$\int_{-\infty}^\infty f(x) \, dx$$
I can simplify the above integral as 
$$\int_{-\infty}^{\infty}f(x)\,dx = \int_{-\infty}^a f(x)\,dx + \int_a^\infty f(x)\,dx $$
$$\lim_{t\rightarrow -\infty} \int_t^c f(x) \, dx + \lim_{w\rightarrow \infty} \int_c^w f(x)\,dx$$
What happens if I get $ -\infty + \infty$ ? 
My professor answered that we cre unable to know where the integral convergences. Is that really the case or not?  

For improper Riemann integrals where any of the limits involved are $\infty$, we say that the integral does not converge, or diverges.  So, for example, $\int_{-\infty}^\infty x\,dx$ does not converge since 
$\lim_{a \to \infty} \int_0^a x\,dx = \infty$ and since
$\lim_{b \to -\infty} \int_b^0 x\,dx = -\infty$.
Note in this case that we may evaluate the limit $\lim_{a \to \infty}\int_{-a}^a x\,dx = 0$.  Although the integral $\int_{-\infty}^\infty x\,dx$ diverges, we state that its Cauchy principal value is $0$.
