$|a\alpha-b\beta|\leq|a-b|+|\alpha-\beta|$
An inequality involving 4 real or complex numbers
How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$? I'm having a trouble with this integral expression:  $$\int_0^{2\pi} \frac{d\theta}{A+B \cos\theta}$$  I've done this substitution: $t= \tan(\theta/2)$  and get: $\displaystyle \cos\theta= \frac{1-t^2}{1+t^2}$ and $\displaystyle d\theta=\frac{2}{1+t^2}dt$ where $\displaystyle \cos^2\theta/2=\frac{1}{1+t^2}$  then the integral becomes:  $$\int\frac{2 \, dt}{(A-B)t^2+(A+B)}= \sqrt\frac{A+B}{A-B} \arctan \left(\left(\sqrt\frac{A+B}{A-B} \right) t\right)$$   However, I'm not sure about the new limits since $\tan$ has period $\pi$ so what I have to do at this point to decide the new limits? And of course if you find some mistake in what I've done before, please let me know! 17.10 Super key to open applications Prove that $x-1$ divides $x^n-1$ In algebra &amp; polynomials, how do we prove that $$x-1 \mid x^n -1?$$ How to unlock Sweet Tooth in Rocket League? How do you get Sweeth Tooth's ice cream van (from Twisted Metal) in Rocket League? Convert LaTeX to HTML Is there a converter that will convert LaTeX to HTML? I have tried several services but the LaTeX diagrams and some of the maths came out weird.  I actually converted the Latex to PDF without a problem. Then I used  to convert from PDF to HTML. The square root signs did not appear properly and some of the Cartesian coordinate plane diagrams had nothing on them except the grid.   \documentclass[12pt,a4paper]{report} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{pstricks,pst-plot} \usepackage{pst-eucl} \addtolength{\hoffset}{0.2cm} \addtolength{\textwidth}{1.1cm} \addtolength{\textheight}{1.0cm} \addtolength{\voffset}{-0.7cm} \linespread{1.3} \begin{document}  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$     $\hspace{30 mm} $     (6)\\  (4) and (6) tell us that\\  (4)  $-1 \leq x \leq 0$  \\  (6)  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$\\  Therefore $-1 \leq x \leq -\sqrt{\frac{1}{2}}$\\  \vspace{50 mm}  \begin{align*} \psgrid[unit=2cm](0,0)(-2,-2)(2,2) \pnode(-1.8,.87){b}\uput[50](b){b} \pnode(-1.8,1.8){a}\uput[50](a){a} \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,4) \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,-4) \psset{linecolor=blue} \psellipticarc(0,0)(2,2){0}{180} \psset{linecolor=red} \psline[linewidth=1pt,linestyle=dashed,dash=2pt](-2,2)(-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt](-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(4,-4) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(-2,2)(-4,4) \end{align*}  \end{document}   PDF output:    PDF->HTML output: Is GTA Online cross-platform? What tool do you use to monitor your servers? When two phrases are pronounced alike but have different spelling and meaning, can we call them homophones?  e.g. "ice-cream" and "I scream", "nitrate" and "night rate", "that's tough" and "that stuff".  Or is there another term for them?  What linguistic phenomenon distiguishes these near homophones?  I've checked the putative duplicate at  but it isn't exactly what I'm asking here. Meaning of Sun Lamp Size? In Cycles, a sun lamp is treated as infinitely far away - only the angle matters, not the position. Other lamps have their sizes defined in blender units: a point lamp with size 1 emits light from a area with size of 1. But if a sun lamp at infinity had a size of 1 blender unit, the angular arc of that sun would be nonexistent.  Clearly, when you change the size of a sun lamp, shadows get fuzzier or sharper. But what does that actually mean? What are you changing? Are there any cheap, fast lenses capable of highschool sports photography? I recently took on my first paying client, and she is also into photgraphy. She usually does sports for her highschool yearbook. They just have an entry level Canon DSLR with a kit zoom on it that goes up to 200mm, but it's not very fast (f/4 or 5.6 or so). She told me that she has a tough time - especially at basketball games - getting shots with any kind of reasonable lighting, and has to do a lot of work in post just to get them reasonable.   mentions the Cannon EF 300mm f/2.8L IS USM II for professional sports photographers. At $6k USD I suspect the yearbook won't be springing for that lens anytime soon.  So I was curious, are there any reasonable quality, fast lenses for Canon cameras that are lower priced (certainly no more than say, $1k)? I'd love to be able to point them to a couple of options. I assume the largest print they'd be making would be a full spread in the yearbook - maybe two 17"x11" (two 8.5" side by side) What is the difference between the bind and live methods in jQuery? Is there a way to fully stretch one line of text on the full width of the page, without manually setting the letter spacing, no matter how many characters on the line? ConcurrentModificationException for ArrayList Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$? USB 2.0 device (scanner) does not work with xhci_hcd on USB 3.0 system Number of combinations of dice having different number of faces that add to $10$ A function $\textbf{face}_{\mathrm{sum}}$ that takes a vector  $f$ that represents the number of faces of each die, and a positive integer  $s$ , and returns the probability that the sum of the top faces observed is  $s$ .  For example, if $f=[3,4,5]$  and  $s≤2$  or $s≥13$ , $\textbf{face}_{\mathrm{sum}}$ returns $0$, and if  $s=3$  or  $s=12$ , it returns  $\frac13*\frac14*\frac15=\frac1{60}$  $\textbf{face}_{\mathrm{sum}}([2,4,4,6], 10)$  I tried to solve this but I can't find the formula to calculate the number of combination of die faces that add to $10$, as the number of faces are different for each die.  Since max no of possible combination is $6*4*4*2=192$, all I need to find is the number of combination of die faces that add to $10$.  Thanks in advance. The future of meta.stackoverflow and meta.stackexchange Historically this site, , has been the place to ask questions about not just Stack Overflow, but the functioning of the entire Stack Exchange (2.0) network, including Area 51 and stackexchange.com and careers.  (Yes, there was a , but that dealt with the old legacy Stack Exchange 1.0 sites. It was deprecated a while ago, and currently redirects here.)  Now that we've gotten many, many months under our belt from the SE 1.0 days, I think it's time that we gave Stack Overflow its own, true per-site meta, and moved the higher level network meta discussions to meta.stackexchange.com.  Over time, it's begun to bug me more and more that requests for Stack Overflow, the site, kind of get buried here on meta.stackoverflow under the avalanche of network issues. Retag requests, synonym requests, things truly specific to Stack Overflow, the site, not the network. This is unfair to our vastly largest and flagship Q&amp;A site, and it's unfair to the Stack Overflow users to mix their requests in with network level concerns.  It's also odd that Stack Overflow is the only network site without a proper, dedicated per-site meta, where its meta has its own reputation system not tied to the parent site in any way.  This is mostly a quirk of history more than anything else, and I believe it is now an appropriate time to split into two sites:   , a proper per-site meta for Stack Overflow with integrated rep , a global network meta with its own reputation system   Yes, this will be painful. But I believe it is a good, necessary, and healthy step both for the future of the network and the future of Stack Overflow.  Some things to discuss:   which questions should be migrated from here to meta.so? Obviously site-specific things like the retag-requests, and perhaps questions with the  tag. Anything that is 100% wholly specific to Stack Overflow and not generalizable to the rest of the network should be moved over. This may be a pretty small, narrow list of questions, and that's OK. Should we do some meta spring cleaning in our transition? Which questions / tags should be blown away as no longer relevant, referring to ancient versions of Stack Overflow or issues that have long since ceased to exist and aren't instructive for any future visitors? Your meta reps will generally be unaffected, except insofar as we delete for spring cleaning, or move site-specific things to meta.so. We still plan to have a distinct reputation system for the new meta.se site, just like here.   I do not have a timeline for this change, that is up to Jarrod and David to decide. It may be months away. However, we feel pretty strongly that this is something that we need to do to clean things up and pave the way for the future -- so I expect it will happen in the next few months, giving us lots of time to decide how we want to do it, per the above.  Update  . Deleting multiple elements from a list Is it possible to delete multiple elements from a list at the same time? If I want to delete elements at index 0 and 2, and try something like del somelist[0], followed by del somelist[2], the second statement will actually delete somelist[3].  I suppose I could always delete the higher numbered elements first but I'm hoping there is a better way. Sum with parameter convergence How do I enable auto-login in LightDM? I want user foo to auto-login using LightDM (which has been used by Ubuntu since version 11.10). How would I do that?