Integral from $\infty$ to $\infty$ I know there is the property that
$$\int_a^a f(x) \mathbb{d} x = 0$$
But what if $a=\infty$? Is the integral still necessarily zero even though infinity isn't well defined?
 A: $\int_a^\infty f(x)dx$ is defined as $\lim_{b\to\infty}\int_a^b f(x)dx$.
$\int_{-\infty}^b f(x)dx$ is defined as $\lim_{a\to-\infty}\int_a^b f(x)dx$.
$\int_{-\infty}^\infty f(x)dx$ is defined as  $\lim_{a\to-\infty,b\to\infty}\int_a^b f(x)dx$.
These are conventional meanings, that everybody agrees on.
But $\int_\infty^\infty f(x)dx$ is not defined to mean anything at all. It is literally meaningless.
A: The expression $\int_{\infty}^{\infty}f(x) dx$ literally means nothing. Still according to conventional ways, you could say it like this :
$$\int_{\infty}^{\infty}f(x) dx =\lim_\limits{a\to\infty}\int_{a}^{a}f(x) dx =\lim_\limits{a\to\infty} 0 = 0  $$
A: Value of your integral depends on a definition of $\int_{\infty}^{\infty}$, for example, take
$$\int_{\infty}^{\infty}f(x)\,dx=\lim_{a\to\infty}\int_{a}^{a}f(x)\,dx=\lim_{a\to\infty}0=0$$
but if we take some other boundaries, i.e.:
$$\int_{\infty}^{\infty}f(x)\,dx=\lim_{a\to\infty}\int_{\ln(a)}^{a}f(x)\,dx$$
it doesn't have to be equal to $0$. For example for $f(x)=1$:
$$\int_{\infty}^{\infty}1\,dx=\lim_{a\to\infty}(a-\ln(a))=\infty$$
We can define it also as
$$\int_{\infty}^{\infty}f(x)\,dx=\lim_{a\to\infty}\lim_{b\to\infty}\int_a^b f(x)dx$$
and the second limit $\lim\limits_{b\to\infty}\int_a^b f(x)dx$ doesn't have to exist at all.
