Integral from $\infty$ to $\infty$

Question

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?

Answer 1

The expression $\int_{\infty}^{\infty}f(x) dx$ literally means nothing. Still according to conventional ways, you could say it like this :

$$\int_{\infty}^{\infty}f(x) dx =\lim_\limits{a\to\infty}\int_{a}^{a}f(x) dx =\lim_\limits{a\to\infty} 0 = 0 $$

Answer 2

$\int_a^\infty f(x)dx$ 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.

Answer 3

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.