I am attempting to solve the integral $$\int_0^\infty \frac{\cos{(\alpha x)}-\cos{(\beta x)}}{x}\,dx$$ I know that the correct answer is $\ln{\frac{\beta}{\alpha}}$ and that the integrand can be rewritten as $$\int_\alpha^\beta \sin{(xy)}\,dy$$ to form the double integral $$\int_0^\infty \int_\alpha^\beta \sin{(xy)}\,dy\,dx$$ Now if we were integrating over a finite rectangular region, I would feel comfortable using Fubini's theorem to swap the order of integration. Is there a way to switch the order of integration for a double integral of this form?
Suppose $ \alpha, \beta&gt;0 $. Compute:   $$ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $$   Here is what I do: $$\begin{align} \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx &amp;= \int_{0}^{\infty}dx\int_{\alpha}^{\beta}\sin (yx)dy\\ &amp;=\int_{\alpha}^{\beta}dy\int_{0}^{\infty}\sin(yx)dx\\ &amp; \\ &amp;   \qquad\text{let $ yx=u $}\\ &amp; \\ &amp;=\int_{\alpha}^{\beta}\frac{1}{y}dy\int_0^{\infty}\sin u du\\ &amp;=\int_{\alpha}^{\beta}\frac{1}{y}dy\left( -\cos u|_{\infty}+\cos u|_0 \right)\\ &amp;=\log\frac{\beta}{\alpha}(-\cos(\infty)+1)\\ &amp;=\log\frac{\beta}{\alpha}-\cos(\infty)\log\frac{\beta}{\alpha} \end{align}$$  But $ \cos(\infty) $ does not exist right? Does it mean the integral actually diverse?  Edit: The question comes from   Who can point out my mistake in the above deduction?
Is it worth getting a suspension seat post as a cheap alternative to full suspension bike? I'm planning a ride from Birmingham to London on the Grand Union Canal (UK), which will be quite bumpy for most of the 145 mile distance.  I have a front suspension bike and can't afford a new full suspension bike.  Do suspension seat posts actually work?  If so, I'll get one of them.  Can anyone recommend a good model? Do I need a US visa to transit (or layover) through an American airport? In Europe usually you don't need to go through immigration to change planes (unless you are entering/exiting Schengen). Is it the case in the US as well? May I catch a connecting flight on one of the US airports if I don't have US visa? Can sudo be reinstalled after being removed? This seems to be a chicken-egg problem. The most common task using sudo is installing and removing software.  sudo apt-get purge &lt;appname&gt;   But sudo itself can be removed.  sudo apt-get purge sudo # Do not run this command on production computers!   This is where the fun comes  ubuntu@ubuntu:~$ sudo bash: /usr/bin/sudo: No such file or directory   Although it's obvious that no person in his right mind will purge sudo (other than me), someone can be fooled to run this command (not directly, in its hex mode, or whatever it's called) or a person could SSH in disguised as tech guru and do the mess.  So is there a way of reinstalling sudo? How can I tell if an iOS device has a GPS? Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq u(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.  I think i should use Liouville's theorem, but how can i do that? Help! How to truncate milliseconds off of a .NET DateTime I am trying to connect to a Linode (running Ubuntu 12.04 LTS) from my local machine (also running Ubuntu 12.04 LTS)  I have created a private and public key on my local machine and copied my public key to my Linode's authorized_keys file.  However, whenever I try to ssh to my Linode I get the error message Permission denied (publickey).    It's not a problem with how ssh is set up on my Linode because I can ssh to it from my Windows machine using key authentication.    In my .ssh directory on my local Ubuntu machine, I have my id_rsa and id_rsa.pub files.  Do I need to create an authorized_keys file on my local machine?    EDIT:  This is what I get when I run ssh -vvv -i id_rsa [youruser]@[yourLinode]:  debug3: authmethod_lookup publickey debug3: remaining preferred: keyboard-interactive,password debug3: authmethod_is_enabled publickey debug1: Next authentication method: publickey debug1: Offering RSA public key: id_rsa debug3: send_pubkey_test debug2: we sent a publickey packet, wait for reply debug1: Authentications that can continue: publickey debug2: we did not send a packet, disable method debug1: No more authentication methods to try. Permission denied (publickey). iOS StackExchange app doesn't handle non-ASCII correctly The iOS StackExchange app doesn't handle non-ASCII characters correctly, see city name in the screenshot below. What is the effect of having correlated predictors in a multiple regression model? I learned in my linear models class that if two predictors are correlated and both are included in a model, one will be insignificant. For example, assume the size of a house and the number of bedrooms are correlated. When predicting the cost of a house using these two predictors, one of them can be dropped because they are both providing a lot of the same information. Intuitively, this makes sense, but I have a some more technical questions:   How does this effect manifest itself in p-values of the regression coefficients when including only one or including both predictors in the model? How does the variance of the regression coefficients get affected by including both predictors in the model or just having one? How do I know which predictor the model will choose to be less significant? How does including only one or including both predictors change the value/variance of my forecasted cost? Temperature and kinetic energy of molecules I was wondering if temperature is related to the average translational kinetic energy of the molecules then why does the average kinetic energy of say a moving object not affect the temperature? What is the mechanism regarding these vibrational motions that imparts temperature. Also do a gas and liquid at the same temperature possess the same kinetic energy? and a heavier molecule traveling at the same speed as a lighter molecule will have a greater temperature? How can I remove fonts that I never use, from libreoffice and linux in general? There are a lot of extra fonts that all seem to look alike and just annoy me in the drop down list. Is there anyway I can erase them from the system? Windows 7 SP1 Windows Update stuck checking for updates How can I install Steam? How does Cypher exit/enter the Matrix without an operator? In The Matrix, Cypher enters the Matrix to speak with the Agents, then leaves again. We never see if he has an operator, and as he kills Tank and Dozer you might assume he doesn't have one.  If he didn't have an operator how does he jack in/out of the Matrix? How can I change what keys on my keyboard do? (How can I create custom keyboard commands/shortcuts?) I want to change the key bindings on some of the keys on my keyboard.  I want some to run commands and others to activate different keys.  What applications can I use to do this?      Index:       (from within System Settings)             I'm still looking for answers using dconf and any other method. How To Change The Fonts In Windows 8 In Windows 7, there is an option to change the fonts for things such as the active title bar, menu, etc. It was found by opening the Window Color and Appearance settings in the Control Panel. I just installed Windows 8 and would like to change the fonts, but I can't find anything to change the actual fonts (there is however a limited drop-down list to change a few font sizes in the appearance settings). Does anyone how to change the fonts used in Windows 8? What type of lens will auto-focus on Nikon bodies without a focus motor? A friend of mine misplaced his passport while staying in Jeju island. Normally I would advise him to just travel to Seoul, get an emergency passport from his embassy and fly back home without issues, but I know that Jeju island has a special immigration regime.   Are there any immigration checks on the way from Jeju island to Seoul? If not, can one travel that route without a passport? How to integrate mv command after find command? I am searching for files which name which contain AAA within their path using following command:  find path_A -name "*AAA*"   Given the output showed by the above command, I want to move those files into another path, say path_B. Instead of moving those file one by one, can I  optimize the command by moving those files right after the find command? How hard do early-career academics in the United States work, really? I'm finishing a PhD in Sweden this year and seeking a post-doc in the United States (atmospheric remote sensing).  It is clear that a job in academia is not a nine to five job; nor do I want it to be.  I often work late in the evening when I'm on a productive spur.  When an important dead-line is coming up, one needs to work harder, and there is no overtime paid.  I accept that.  However, no matter how much I'm interested in my research, I do enjoy and need a reasonable amount of spare time, too; relaxing on the weekends, occasionally a long weekend away, and sometimes a longer trip, such as three weeks in the wilderness.  Regarding the normal work ethos for early-career academics in the United States, I have a hard time judging what is normal and what is excessive.  Some examples:    warning post-docs that he expects all of the members of the group to work evenings and weekends. My apparent naïveté in believing that , despite Wikipedia  as a rest from work, or a hiatus, often lasting from two months to a year.  I was thinking of my friend, who spent a year between his PhD and his first post-doc travelling from France to Mongolia mostly on foot. Someone  I am not a good role model, but when I don't work on a Sunday, I count that as a vacation day.  having no employment-related benefits such as paid vacations, sick leave, or unemployment compensation. Question    We don't have such a work-ethos where I'm at.  I belief that working too hard risks stress and burn-out, and does not increase productivity in the long run, nor human well-being.  I want to do science.  Doing science makes me happy, but having time to relax while not doing science is important for me.  Does the selection of examples I gave above represent a normal situation in U.S. early-career academia?  Should I expect an attitude where asking for a 3-week holiday during the summer is considered as being not serious, or is the situation in practice usually not as bad as the examples above make it sound?  How hard to early-career academics in the United States work, really?