1. ## Integral question

Hi

I want to integrate the following:

1/(ln(x)^3) from 2 to infinity.

Am i allowed to do:

(ln(x)^-3)
-3 ln(x)
pull out -3 and integrate ln (x)?

2. Originally Posted by Kuma
Hi

I want to integrate the following:

1/(ln(x)^3) from 2 to infinity.

Am i allowed to do:

(ln(x)^-3)
-3 ln(x)
pull out -3 and integrate ln (x)?

Of course you're allowed, but be sure the power is on x and NOT on the whole ln(x)...

Tonio

3. Originally Posted by Kuma
1/(ln(x)^3) from 2 to infinity.

Kuma: even if you don't use $\LaTeX$code, you can avoid the ambiguous notation writing:

1 / [ ln (x^3) ] or 1 / [ ( ln x )^3 ]

which are different functions .

4. Ah its the entire [ln(x)]^3.
What do i do in this case? Substitution?

5. $
\displaystyle
\int \frac{dx}{ln^3(x)}=\int \frac{x \ dx}{xln^3(x)}=\int \frac{x \ d(lnx)}{ln^3(x)}=\int x \ \frac{(-1)}{2} \ d \frac{1}{ln^2(x)}=
$

$
\displaystyle
=-\frac{1}{2} \ \int x \ d \frac{1}{ln^2(x)}
$

You may use substitution (I usually don't use it)

$
\displaystyle
u=x ,\ \ v=\frac{1}{ln^2(x)}
$

We get

$
\displaystyle
=-\frac{1}{2} \ [ \ x \ \frac{1}{ln^2(x)}-\int \frac{1}{ln^2(x)} \ dx \ ]
$

You may continue in the same way.

Also we may notice x>>1

$
\displaystyle
\frac{1}{ln^3(x)}> \frac{1}{x} \ .
$