# Thread: Condition of an exponent for convergence of an integral

1. ## Condition of an exponent for convergence of an integral

I must find for which values of $p \in \mathbb{R}$ the following integral converges : $\int_e^{+\infty} \frac{dx}{x\ln^p|x|}$. Mathstud28 already done that (well, something pretty similar) in one of my earlier threads, but I didn't understand a step he did, so I would like a little bit more explanations :
Hmm, sorry the quote doesn't work. I copy and past it, but it only shows the first line and all the other is cut, so I quote it by parts :[quote].
Still doesn't work!
Anyway, he was working with $\int_2^{+\infty}\frac{dx}{x\ln^p|x|}$. He said that if $0, then the indefinite integral is equal to $\frac{\ln^{-p+1}(x)}{-p+1}$. This is what I missed to understand. Did he do integration by parts? How could he reach this result? Thanks!!

2. Still doesn't work!
Anyway, he was working with $\int_2^{+\infty}\frac{dx}{x\ln^p|x|}$. He said that if $0, then the indefinite integral is equal to $\frac{\ln^{-p+1}(x)}{-p+1}$. This is what I missed to understand. Did he do integration by parts? How could he reach this result? Thanks!!
If $p\ne{-1}$

Then $\int\frac{dx}{x\ln^p(x)}=\int\frac{\frac{dx}{x}}{\ ln^p(x)}$

Now let $u=\ln(x)$

3. This is mathstud's baby, so I will leave him answer. I am sure he is typing as I write this. Anyway, here is a site with a list of integrals you may find helpful instead of deriving them each time. Unless you want to.

Definite Integrals, General Formulas Involving Definite Integrals

4. Thanks mathstud28, I reached it! I thought it would have been much more complicated.

5. Originally Posted by galactus
I am sure he is typing as I write this.
Or finishing before you
Originally Posted by arbolis
Thanks mathstud28, I reached it! I thought it would have been much more complicated.
Yeah, sometimes the obvious alludes even the best of us.

6. Hey mathstud, you know what might be fun?. Go to the link I posted above and actually derive the solutions they give for the integrals. You have probably done that already.

Some of them are probably very challenging.