Integral: Convergence/Divergence?

[golriz]

Homework Statement

hi
i have a question please help me:
does the following integral converge or diverge?
∫_2^∞▒ln⁡〖x 〗/(e^x+1) dx
thanks

Homework Equations

The Attempt at a Solution

 

[Dick]

What have you tried?

 

[HallsofIvy]

First, as Dick said, we need to know what you know and what you have tried before we can make any suggestions. Second, you seem to have used several "special" characters that do not show up on my internet reader. Could you please restate the integral without them?

 

[golriz]

oh i have showed you my question.
my question is:
"Use the coparison test to determine if the following integral converge or diverge:
integral of [ ln x/ e^x + 1 ]dx from 2 to infinity "
i don't know can you understand my question now?

 

[Dick]

I understood the question to begin with. The question was do you understand it. If you are going to use a comparison test you first have to take a guess as to whether it converges or diverges. What's your opinion? If you think it converges find another function that's greater than the given function and show it converges. If you think it diverges find another function that's less than the given function and show it diverges. What's your choice?

 

[golriz]

i know the comparison test but i don't know the solution of this problem.
because i should choose a function greater or less than the given function that converging or diverging of these functions be obvious for me, so i don't know these functions.
do you understand my mean?

 

[Dick]

I understand your meaning. I'm just asking for your opinion on how to start. ln(x) grows really slowly, e^x grows awfully fast. Do you guess it will converge or diverge? I.e. should we look for a lower bound or an upper bound?

 

[golriz]

i think it converges and we should look for upper bound

 

[Dick]

Good, I agree. So we have to look for something larger than ln(x) to put in the numerator and something smaller than e^x+1 to put in the denominator. The resulting integral should be something that i) we can integrate and ii) converges. Any guesses?

 

[golriz]

i'm not sure but can i suggest x/e^x we can integrate it by parts

 

[Dick]

i'm not sure but can i suggest x/e^x we can integrate it by parts

That's exactly what I would have suggested. It converges and it's larger than ln(x)/(e^x+1). Well done.

 

[golriz]

oh really thank u so much because of your help
so no need to proving the converging of x/e^x?

 

[Dick]

Yes, you have to prove it. Don't be silly. I know it converges because I've done this stuff before. If the person who is grading your work knows it does and knows that you know it does and you have an agreement that you don't have to show it, then you don't. It's not hard is it?

 

[golriz]

no not hard
thanks a lot again
bye

 

[Dick]

No problem. You are pretty good at this. You take hints well. I didn't really tell you much you didn't already know. Bye.