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Adding the consensus to review's history A curious integral The following integral has been on my mind for a while $$\int_0^\infty \frac{\sin(x)}{e^x-1}\,\mathrm d x \tag{$\dagger$}$$    Let us indicate the integrand as $f(x)=\frac{\sin(x)}{e^x-1}$. The following are a couple of observations.   The integrand can be extended by continuity in $0$ since $$\lim_{x\to 0}f(x)=\lim_{x\to 0}\frac{\frac{\sin(x)}{x}}{\frac{e^x-1}{x}}=1$$ This is the original reason I started playing around with this integral. Mathematica yields the result $$\int_0^\infty \frac{\sin(x)}{e^x-1}\,\mathrm d x=\frac{\pi}{2}\textrm{Coth}(\pi)-\frac 12 \approx 1.076674047$$ which, following numerical evidence, seems correct. Complex analysis may be useful here, since the integrand is a holomorphic function on $\mathbb C$. I tried writing $\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$, but I was not able to find an appropriate integration contour to solve the problem. The $-1$ in the denominator breaks the simmetry of the expression. This made most of my substitutions useless.   Can this integral be evaluated correctly, preferrably through complex analytic methods? MySQL: SELECT from another server I'm afraid that I already know the answer to my question, but I'll ask it anyway:  When there are two MySQL DB servers, can I access data that is stored on the other server?   In other words: Can I somehow do this:  INSERT INTO table (x, y, z)    SELECT x, y, x+y       FROM [otherserver].[database].[table]   Is the answer really as short as "No"? I want to update Acrobat Reader for Firefox, but the download has the .bin extension. How can I install it? 90s (or early 2000s) scifi (mini)series where a boy and old man chase a black guy in metal suit Does MariaDB or MySQL implement the VALUES (expression) table value constructor? It is often stated that the Lagrangian formalism and the Hamiltonian formalism are equivalent.   We often hear people talk about eigenvalues of Hamiltonians but I have never ever heard a word about eigenvalues of Lagrangians.   Why is this so? Is it not useful? is it not possible to do it? Hi is it possible to get images from go daddy  to visual force pages.if it possible can you explain clearly how the process is. I thought people would be working on little code projects together, but I don't see them, so here's an easy one:  Code that validates a valid US Zip Code. I know there are ZIP code databases out there, but there are still uses, like web pages, quick validation, and also the fact that zip codes keep getting issued, so you might want to use weak validation.  I wrote a little bit about zip codes in a side project on my wiki/blog:    There is also a new, weird type of zip code.     I can do the javascript code, but it would be interesting to see how many languages we can get here. Why are the even and odd Regge trajectories degenerate? How to download image using requests Why do objects in front and back view look reversed? Whenever I model objects that has different look at front and back. Viewing it from front perspective or orthographic the model displays back of the object. I just started real analysis.  I don't have a background in proofs or logic, simply calculus.  So I'm trying to learn more about proofs--so forgive the basic question, please.  How do you go about proving this theorem: If $a$ and $b$ are any numbers, then there is one and only one number $x$ such that $a + x=b$.  This number is given by $x=b+(-a)$.  First part of the proof I understand--it's simple.  We simply use some axioms and do the following:  $a+b-a=b$, which of course is true.  This is just true from plugging in the $b+(-a)$ for $x$.  But how about the uniqueness issue?  In my text, it says:  $$(a+x)+(-a)=b+(-a) x=b+(-a)$$  How does this prove uniqueness?  Is this equation essentially saying that the ONLY possible value of $x$ is expressed by $b+(-a)$?  How would I have known to set this up as part of the proof if I was asked to prove something is unique?  Sorry for the basic question--have to start somewhere. Whenever I start a game, from both the disc and hard drive, my Xbox will say "Username signed out". What's wrong and how do I fix it? Just to be clear, it signs me out when I start a game. How can I stop this? I have heard from  that stable orbits (ones that require a large amount of force to push it significantly out of it's elliptical path) can only exist in a three spatial dimensions because gravity would operate differently in a two or four dimensional space. Why is this? Difference between  and I have an interesting mathematical/electrical dilemma and I'm not sure where to start.  I am considering buying  (specs on page).  I will be hopefully buying a battery pack with it to support a long-road trip (a few months) wherein I'll keep medication in the fridge (powered by the battery pack when parked and by the car when driving). The medication needs to be keep between 2 and 8 degrees C at all times.  Here's the  I'm thinking of getting:  Can someone please help me understand what I need to do in order to calculate whether or not this fridge + battery combo will work, and if so, how much time will I be able to power the fridge for.  Ideally, I need a combination of fridge and battery which will allow me to have the fridge constantly powered, assuming I sleep in the car for approx. 8 hours a night and drive for the rest of the day (although, there will be times which I'm away from the car during the day for several hours + times where I hope to have access to a wall outlet).   Specs:   Fridge:   Input voltage (AC 220-240 V Input frequency 50/60 Hz Rated input power (AC) 64 W Rated input power (DC) 46W DC12V/50W DC24V W Input voltage (DC) 12/24 V   Battery:   Capacity: 453Wh Battery Type: Lithium-ion Input Voltage: AC 110-220 V, 50/60 Hz; Solar Power: 18 V; Car Input: 12 V Output Voltage: AC 110V 60 Hz; DC Output: 4 * 12V 4 A (Max.10 A); USB Output: 4 ports: 2* 5V 2.1 A and 2* 5V 1 A Power Output: Rate 500 W Peak/Surge 1000 W Does RSA work for any message M? I am rebuilding an old server.  Only one of the five old sites need to go live and all five of the old sites point to the same IP address.  Unfortunately any time a request is made for one of the old sites via their own unique domain names they are passed the pages from the one site that I'm trying to make live.  I would just like the old sites to display a 404 instead of the one site that I'm actually trying to make live.  As an example:   Site 1 (192.168.0.1) [www.first.com] Site 2 (192.168.0.1) [www.second.com] Site 3 (192.168.0.1) [www.third.com]   I want to move site 1 live again.  So I turned on apache on the machine with the IP address of 192.168.0.1, with all of site 1's files in the /var/www directory.  When I browse to 192.168.0.1 or www.first.com I correctly see site 1. When I browse to www.second.com and www.third.com I also receive site 1 which is the problem.  Is there a way I can 404 any request which comes in without www.first.com?  Here is my sites-available:  NameVirtualHost *:80  LogFormat "%h %l %u %t \"%r\" %&gt;s %b" common  &lt;Files ~ "^\.ht"&gt;     Order allow,deny     Deny from all     Satisfy All &lt;/Files&gt;  &lt;Directory /&gt;   Order Deny,Allow   Deny from all   Options None   AllowOverride None   Options None &lt;/Directory&gt;  &lt;Directory /var/www&gt;   Order Allow,Deny   Allow from all &lt;/Directory&gt;  &lt;VirtualHost *:80&gt;         ServerName first.com         ServerAlias www.first.com         DocumentRoot /var/www         ErrorLog /var/log/first.com.err         CustomLog /var/log/first.com.log common &lt;/VirtualHost&gt;  &lt;VirtualHost _default_:*&gt;         ServerName unknown         DocumentRoot /var/www_404 &lt;/VirtualHost&gt; Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.   The hint I was given is to evaluate: $$E[X-Y;X&gt;a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$  which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X&gt;a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.  But I need some more hints.