Question

\[integrate \int\limits_{e}^{e ^{2}}\frac{dx}{x(\ln x)^{2}}\]

Answer 1

\[\int\limits_{e}^{e ^{2}}\frac{dx}{x(\ln x)^{2}}\]
Put
\[\ln x=t\]
so we'll get
\[\frac{1}{x} dx =dt\]
Can you change the limits?

Answer 2

yup.
u = lnx
if x = e^2 ; => 2
x = e ; => 1

Answer 3

Great :D
so we have now
\[\int_{1} ^{2} \frac{dt}{t^2}\]
I think you can do now, can't you?

Answer 4

i don't think so.
but i know the answer is 1/2 but i don't know the solution. can you please show me?

Answer 5

We know integral of
\[\int x^{n} dx=\frac {x^{n+1}}{n+1} where\ n\ne -1\]
here we have
\[\int_{1}^{2} \frac{dt}{t^2}=\frac{t^{-2+1}}{-2+1}\]
on simplification we get
\[\int_{1}^{2} \frac{dt}{t^2}=[\frac{-1}{t}]_1^{2}\]
Now just apply the limits

Answer 6

Do you understand this?

Answer 7

yup. Thanks a lot @ash2326 . :)

Answer 8

You're welcome:D