Improper Integral : Convergent and divergent

[flat kestrel]

Can someone please help me with no 2. How do i prove that it is divergent

[inner crater]

Find an integral that is strictly less, and show that that intergal diverges

[flat kestrel]

yeah but i dont know what I should use as the g(x) that is less than that but is integratable

[inner crater]

For $0\le x\le 1$, can you think of a constant upper bound for $\ln^2(x+1)$?

[safe lotus]

SWR

[flat kestrel]

Upper bound: (ln2)^2
Lower bound : 0
?

[inner crater]

yeah that's fine. Personally, I'd make the upper bound a nice integer like 4, but yours is good too

[flat kestrel]

then what do you do with it?

[inner crater]

oh hold on

Yeah we wanted a lowe bound, not upper

[flat kestrel]

yeah

[inner crater]

okay that is trickier

oh

oh

I think I know

(maybe?)

[flat kestrel]

i tried using 1/x^4 but it doesnt work becuase it is higher

[flat kestrel]

how?

[inner crater]

maybe try splitting it into $\left(\frac{\ln(x+1)}{x}\right)^2\left(\frac{\sqrt[3]{x^2+x\sin x}}{x^2}\right)$

[safe lotus]

SWR

[inner crater]

I think (ln x)/x has a nice bound

,w plot (ln (x+1))/x

[safe lotus]

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Requested by spelwerdzrite

[flat kestrel]

oo alright lemme try that

[inner crater]

yeah that is bound below by 0.5

(idk how hard it is to prove that)

anyway I am off

hopefully that helps