Definite integral

[noble sentinel]

Brain stuck error 404 help

[craggy pulsar]

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[noble sentinel]

I tried differentiating under the integral sign with x^n-x^(n+2) as the numerator

and then I'm stuck

[lapis canopy]

integrand is unbounded

break into two parts and see if either even converges

@noble sentinel

[noble sentinel]

It's unbounded but I'm not sure what to use to break it

I think desmos shows asymptotes at ln2 but I'm not sure how to arrive at that logically

[lapis canopy]

what's the problematic point in (0,1)?

[noble sentinel]

Somewhere at 0.609

[lapis canopy]

mhm

(sqrt(5)-1) /2

[noble sentinel]

I guess that's the x^2+x-1 term being 0?

oh right yeah

[lapis canopy]

what happens with

$$ \int _0^{(sqrt(5)-1)/2}\frac{x^2-x^4}{(\ln x)(x^2+x-1)}dx $$

[burnt nimbus]

aL

[lapis canopy]

does this converge?

[noble sentinel]

Nope

Atleast not according to desmos

[lapis canopy]

that is correct, you can justify it via comparison as well

hence, the initial integral diverges

[noble sentinel]

It diverges on both sides right?

[lapis canopy]

didn't check the other half, but I'd assume so

[noble sentinel]

Is there a way to subtract the terms off?

[lapis canopy]

wdym

[noble sentinel]

Logically, you could somehow cancel out the positive and negative unbounded areas since they're symmetric around x= sqrt5-1 /2

and end up with a finite area

[lapis canopy]

no

the integral is improper, if either half diverges that's it

not integrable

[noble sentinel]

Alright, thank you then

[lapis canopy]

🍺

you could ask similarly if

1/x from -1 to 1 converges

the answer is no

even tho there is symmetry

[noble sentinel]

right that's true

+close