Question

Integration: see first post

Answer 1

\[\int\limits \frac{ 1 }{ x(lnx)^{2} } dx \]

Answer 2

There's a substitution just begging to be made.

Answer 3

Thats what i was thinking lnx=t so dt=1/x ? so then
\[\int\limits \frac{ 1 }{ t^2}dt \]
and then
would result in:
\[-\frac{ 1}{ 3 } \frac{ 1 }{ t ^{-3} }\]

Answer 4

Derivative of lnx is 1/x... maybe that'll help ;)

Answer 5

A bit off... either it has a positive exponent and it's in the denominator, or it has a negative exponent and it's in the numerator...
negative exponent in the denominator means something else entirely -_-


Correct substitution, though.

Answer 6

Wait a sec... it's not just a bit off... it's way off XD
\[\Large \int t^n dt = \frac{t^{n+1}}{n+1}+C \qquad n\ne -1\]

Answer 7

ah yea i went the wrong way i feel dumb haha to early for this

let me try that again:
\[-1t ^{-1}\]

Answer 8

That's right :D
Now switch back that t.

Answer 9

so then to simplify:
\[\frac{ -1 }{ \ln x }\]

thank you thank you letting the series stuff rattle me im forgetting basics.

Answer 10

Series? D:

Answer 11

using this for the integral test for convergence divergence

Answer 12

I see... sounds like fun XD
\[\Large \sum_{n=1}^\infty\frac{1}{n[\ln(n)]^2}\]

Answer 13

Oh wait, it can't start at 1...
what was I thinking, LOL
\[\Large \sum_{n=2}^\infty\frac{1}{n[\ln(n)]^2}\]

Answer 14

i was just about to ask about that i had wrote 1 down on my paper as well... the book had 2

Answer 15

had *written* :>

Anyway, details details...

Why don't you share some more convergence-divergence questions, those are entertaining XD

Answer 16

this would be convergent right?

Answer 17

evaluate it from 2 to infinity XD

Answer 18

ok so what i had:
\[\lim_{t \rightarrow \infty} (-\frac{ 1 }{ lnt }+\frac{ 1 }{ \ln2 })\]

would make the first term go to 0? or is that wrong

Answer 19

First term goes to zero indeed :)

Answer 20

then the intergral converges at:
\[\frac{ 1 }{ \ln2 }\]

which means the original equation is convergent?
if i read the rule correctly

Answer 21

yup.

Answer 22

thanks a ton