Antiderivative of 1/((x(lnx)^3) using u-substitution

[stau40]

Homework Statement

I'm working on an infinite series problem and need to find the antiderivative of 1/((x(lnx)^3).

Homework Equations

u=lnx

The Attempt at a Solution

I know I have to use the substitution u=lnx, but I still can't figure out what the answer is. I know the antiderivative of 1/((x(lnx)) is ln(lnx) but the third power in my problem is giving me trouble. Any advice? Thanks!

 

[Bohrok]

After the u-substitution, what is the integrand in terms of u now?

 

[stau40]

It would be 1/(x(u)^3)

 

[Mark44]

It would be 1/(x(u)^3)

When you make a substitution, replace everything. Here you still have a factor of x remaining. If u = ln(x), what is x in terms of u? Also, and this is related, did you replace dx by its appropriate expression involving du?

 

[stau40]

Ok, it should be the antiderivative of du/(u^3) but I still don't understand how to work thru the third power. If I solve for the antiderivative and end up with u^4 I would need (1/4) in front of u and that gets me to the wrong result.

 

[Mark44]

Ok, it should be the antiderivative of du/(u^3) but I still don't understand how to work thru the third power. If I solve for the antiderivative and end up with u^4 I would need (1/4) in front of u and that gets me to the wrong result.

Well, yes. You can't go from du/u^3 to u^4.

du/u^3 = u^(-3)du

 

[stau40]

The light finally came on, Thanks!