Improper integral divergent convergent

[cobalt portal]

Someone can guide me step by step?

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[naive yew]

How is the improper integral defined?
It should be the limit (if exists) of some proper definite integral(s). What would that/those definite integral(s) be?

[cobalt portal]

Is it like this?

But I'm currently stuck

[gloomy glade]

Can't you use feynman's trick in this case?

I am pretty sure it is the example of Feynman's trick

Or a similar integral

[hexed ice]

well for one thing it should integrate to -1/2 e^-x^2
this is not very relevant though
to find those limits, i mean, you are raising e to the power of -infinity

[cobalt portal]

Like this?

[cobalt portal]

I get zero

[hexed ice]

yeah it is definitely 0 by symmetry

i thought the goal was to prove convergence though

but yeah

[cobalt portal]

Wait that's not how to prove it? How to prove it though?

[hexed ice]

splitting it up into the 2 improper integrls with the limits is good

i mean my instinct would be to prove its <= e^-|x| in absolute value or something (which it is)

but this is actually fine in this case

[cobalt portal]

So if I got zero, then the integral is convergence?

[hexed ice]

if you get a finite number it converges

if you split it up, that is

there may be integrals where $\int_{-a}^{a} f(x) dx$ is always $0$, for example, but its wrong to say $\int_{-\infty}^{\infty} f(x) dx = 0$ because $\int_0^{\infty} f(x) dx$ diverges

[hazy tree]

cute rizzly bear (nom nom nom)

[hexed ice]

such as f(x) = x

that's why they define int -inf to inf that way

[cobalt portal]

Owh I see