# Integral with natural log problem

1. Jul 8, 2012

### 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.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited: Jul 8, 2012
2. Jul 8, 2012

### jackmell

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.

3. Jul 8, 2012

### 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.

4. Jul 8, 2012

### luxxx

Well I did u substitution for u = ln x.

5. Jul 8, 2012

### jackmell

That would work. And note:

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

6. Jul 8, 2012

### luxxx

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

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