A community for students.
Here's the question you clicked on:
 0 viewing
JamesJ
 4 years ago
Definite integral. Here's a fun one.
JamesJ
 4 years ago
Definite integral. Here's a fun one.

This Question is Closed

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.5\[ \int_0^\infty \frac{x}{e^x  1} \]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I was working on this one for a bit earlier, pretty sure it isn't solvable by standard methods. I wolframed it to find that the answer is \[\frac{\pi^2}{6}\] so I'm assuming that it is possible to convert it to the form of Basel's problem. I also tried converting it to one of the integrals I know is equal to that, like \[\int_0^1 \int_0^1 \frac{dxdy}{1xy}\] I also found several different Taylor Series for the integrand, but that didn't lead anywhere. I didn't attempt to use complex analysis yet, since I would rather that be my final effort, haha. I have a page and a half of scribbles but nothing that jumps out at me as convertible to Basel's problem. Any particular hints that you would suggest? It seems like a really interesting question and I'd like to solve it.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.5write the integrand as \[ \frac{xe^{x}}{1e^{x}} \] and now write that as a geometric series.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Ah, I see now. I tried to convert it to a geometric series several times, but for some reason never tried that one (even though I have that fraction scribbled in a corner of the page!). Very very nice problem. For sake of completeness: \[\int_0^{\infty}\frac{xdx}{e^x1}=\int_0^{\infty}\frac{xe^{x}dx}{1e^{x}}=\int xe^{x}(1+e^{x}+e^{2x}+ \cdots)dx\] \[=\int x(e^{x}+e^{2x}+\cdots)dx\] \[=x(e^{x}+\frac12e^{2x}+\frac13e^{3x}_0^{\infty} + \cdots)+\int e^{x}+\frac12e^{2x}+\frac13e^{3x} +\cdots dx\] The left goes off to zero, leaving \[e^{x}+\frac14e^{2x}+\frac19e^{3x}+ \cdots _0^{\infty}\] \[=\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}\] Beautiful :)

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.5here's another one... http://openstudy.com/study#/updates/4f0e4b24e4b04f0f8a91269e which points to the solution to the more general problem: \[ \int_0^\infty \frac{x^n}{e^x 1} dx \]
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.