Integral with natural log problem

[luxxx]

Hello,
The problem is ∫(ln x)/(x + x ln x) dx.

I've done most other problems in the set, but don't know where to start with this one. Although we are just learning integration by parts, I'm not sure how this would apply. I can get to ∫u/(1+u) du
Thanks for any help.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[jackmell]

Hello,
The problem is ∫(ln x)/(x + x ln x) dx.

I've done most other problems in the set, but don't know where to start with this one. Although we are just learning integration by parts, I'm not sure how this would apply. I can get to ∫u/(1+u) du
Thanks for any help.

When you have multiple versions of the exp and log function, try changing variables by letting w=e^x or w=ln(x) or x=ln(w) or x=e^w. Try those and see what happens.

 

[luxxx]

I think the answer is 1 + ln x - ln ǀ1 + ln xǀ +C, but that's using an integration table.. I'd like to know how you would get there.

 

[luxxx]

Well I did u substitution for u = ln x.

 

[jackmell]

That would work. And note:

$$\frac{u}{1+u}=1-\frac{1}{1+u}$$

 

[luxxx]

Ha, yeah I just got that when you responded. Thank you!