Definition of improper integral with bounds -∞ and +∞

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[icy eagle]

I'm pretty sure that's just a typo.

[slim stratus]

Ok, glad it's not just me who thinks there is a typo but then that leads to the question of why the definition being used is still different ;-;

[icy eagle]

There aren't two definitions of the same thing. The first definition is the improper integral from a to infinity, the second is the improper integral from minus infinity to infinity.

[slim stratus]

Er, probably my bad for being unclear. Maybe an example would help.

Consider the integral with bounds of -∞ and +∞, as in the definition near the end of the screenshot, where f(x) = sin(x). If the definition in this image is used, then this would converge to 0, right? But if you split the integral and take the limits separately (contrary to this definition) then it would not converge. This second way is what I've previously been taught is how to handle such integrals.

[icy eagle]

Do you mean sin(x)/x? The integral of just sin(x) doesn't converge.

[slim stratus]

Nope, sin(x)

As it's an odd function that does not normally converge but looks like it would with this definition..?

[icy eagle]

Hmm, show your work on the split integral.

[slim stratus]

Sorry, to clarify, you mean showing that when you split the integral to be from -∞ to 0 then 0 to +∞ that it doesn't converge?

Doesn't

That was a typo, mb

[icy eagle]

Yeah.

[slim stratus]

Oki, I can write it out clearly if you'd like but it's just a case of saying that the limit of cos(x) as x --> ∞ is undefined, no? And it will take me a little bit to do 😭

[icy eagle]

Right, but remember, you have a sum of integrals.

The integrals separately might not converge, but their sum might.

[slim stratus]

Hey @icy eagle
Thank you for your help, but I think I'm gonna leave this for now. It is very late for me and I don't think this is going anywhere too useful 🫶