I tried solving this integral:  $$\int^{\pi/2}_{-\pi/2} \frac{1}{2007^x + 1}\frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$  I took a while before aptly applying the following identity I had noted down in my notebook before.. $$\int^m_{-m}\phi(x)dx=\int^m_0\phi(x)+\phi(-x) dx$$  And my hunch was right; I managed to get it to a 'manageable' form: $$I_1=\int^{\pi/2}_0\frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$    This got me wondering..  Does there exist a closed form for the following integrals?  $$ I_2=\int \frac{\sin^{a}x}{\sin^{a}x+\cos^{a}x}dx$$$$ I_3=\int \frac{\cos^{a}x}{\sin^{a}x+\cos^{a}x}dx$$
I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand.  This problem is from some old high school IMO training courses.  I am not sure how to solve it. The only information that may be of help is $$ \sin x=\sum_{n=1}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \quad \cos x=\sum_{n=1}^\infty \frac{ x^{2n}}{(2n)!} \quad \forall \ x $$ Thanks, I am looking for a complete solution, not a description as of what to do.
Using one group separator for integer part and another for decimal part (siunitx) I thought I was able to do something like this before, but I can't find my solution.  I would like to format my large numbers as follows:   12,504.3124 2359 3   I know I can use siunitx to partially set this up, but I'm not getting the results I want:  \documentclass{article} \usepackage{siunitx}  \newcommand\aenumA[1][4]{%%   \num[group-separator={,},        group-digits=integer,        ]}  \newcommand\aenumB[1][4]{%%   \num[group-digits=decimal,        ]}  \pagestyle{empty} \begin{document}    \aenumA{12504.312423593}    \aenumB{12504.312423593}  \end{document}   I would like a hybrid solution (commas in the integer part and spaces in the decimal).  I would actually also like to be able to group differently between the two: every three digits in the integer part, every four digits in the decimal part.  Is there a straight-forward way to do this with siunitx?  Or, do I need to split the number at the decimal and treat the two parts separately and then glue them back together? I am looking for a convenient way to set the background of tabular cells. I'd like to do two things:  (1) Set the background shaded in a color (2) Set the background hatched  I learned that the TikZ package provides means to do this but I am unable to come up with a solution that does not use matrices (such as here: ) but instead uses the tabular package.  Here's what I'd like to do:  \documentclass{article} \begin{document} \begin{tabular}{|l|l|l|} \hline   1 &amp; \setBGGray 2 &amp; 3 \\\hline   4 &amp; 5 &amp; 6 \\\hline   7 &amp; \setBGHatched 8 &amp; 9 \\\hline \end{tabular} \end{document}   The output should look like this:    Can someone please help me achieving this using tabular (or tabularx) and TikZ? Can we have a system to submit potential robo-reviewers for moderators attention? Is it possible to get rid of the bullets in the miniframes outer theme Search All Fields In All Tables For A Specific Value (Oracle) How to connect to Oracle as "SYS" from SQL*Plus in Java My laptop running Windows 7 Ultimate (non OEM) died on me today and it has tonnes of stuff on it - my projects, Visual Studio and office installations, etc. etc.  I am getting a new laptop from somewhere and was wondering if I could just unplug the hard disk from my original (dead) machine and plug it into the new one.  I remember trying this once with Windows Xp (on a desktop) and the OS crashed on boot and I had to reinstall everything again.  I have read about  but by the looks of it, it can be used only when the host computer is alive and kicking -- which mine isn't.  My old laptop was a Compaq Pressario v3000 (3 years old) on an AMD and the new one is going to be a new dual core Intel HP laptop.  Any ideas?  Thanks! When to use "I" and "me" in an answer I can show that for $x &gt; 0$ and $r_{i} &gt; 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ However, I can't do this using straight up induction, strong or weak. Can someone do this?   You can show the inequality holds for $n$ if it holds for $n+1$ using $r_{n+1} = (r_1 \cdots r_n)^{1/n}$, so going down isn't a problem, but going up has eluded me. Additionally, you can show that if it holds for $n=a$ and $n=b$ then it holds for $n=ab$. Using powers of two and the above is sort of a way to prove the above by induction but it certainly isn't "normal".  You can also prove the above using Lagrange multipliers (which isn't surprising). The last proof I know is where you compare coefficients of $x^k$ and use the AM-GM inequality. I'm not a git master, but I have been working with it for some time now, with several different projects.  In each project, I always git clone [repository] and from that point, can always git pull, so long as I don't have outstanding changes, of course.  Recently, I had to revert to a previous branch, and did so with git checkout 4f82a29.  When I was again ready to pull, I found that I had to set my branch back to master.  Now, I can not pull using a straight git pull but instead, have to specify git pull origin master, which is annoying, and indicates to me that I don't fully understand what is going on.  What has changed which does not allow me to do a straight git pull without specifying origin master, and how to I change it back?  UPDATE:  -bash-3.1$ cat config [core]     repositoryformatversion = 0     filemode = true     bare = false     logallrefupdates = true [branch "master"] [remote "origin"]     url = git@github.com:user/project.git     fetch = refs/heads/*:refs/remotes/origin/*   UPDATE 2: To be clear, I understand that my original method may have been incorrect, but I need to fix this repo so that I can simply use git pull again.  Currently, git pull results in:  -bash-3.1$ git pull You asked me to pull without telling me which branch you want to merge with, and 'branch.master.merge' in your configuration file does not tell me either.  Please name which branch you want to merge on the command line and try again (e.g. 'git pull  '). See git-pull(1) for details on the refspec.  If you often merge with the same branch, you may want to configure the following variables in your configuration file:      branch.master.remote =      branch.master.merge =      remote..url =      remote..fetch =   See git-config(1) for details.   I can tell git pull which branch to merge, and it works correctly, but git pull does not work as it did originally before my git checkout. I want to reinstall Ubuntu, but I also want all information and settings to disappear. So everything looks like the first time I installed it. Is it possible to do so via Terminal? Or what are the steps I should follow? Is "Five dollars is all I have" correct? I want to emphasize that I only have 5 dollars, so can I use the sentence:     Five dollars is all I have.   Or must it be:      Five dollars are all I have.   Please give the explanation as well. Is there a correct resistance value for I2C pull-up resistors? Can I run two 4k (3840x2160) displays at 60hz on a Macbook Pro 13" 2015 I think the title says it all. Can I run two 4k (3840x2160) displays, both at 60hz on a Macbook Pro 13' 2015? I understand they might need to be SST and each connected to a Thunderbolt 2 port. But I cannot get a confirmation whether that setup would work.    says:     Simultaneously supports full native resolution on the built-in display and up to 3840 by 2160 pixels on up to two external displays, both at millions of colors.   …   Support for up to 5120-by-2880 resolution at 60Hz on a single external display (model with AMD Radeon R9 M370X only)   Unfortunately, it does not state at what frequency the two displays can be run.    says:     With OS X Yosemite v10.10.3 and later, most single-stream 4K (3840x2160) > displays are supported at 60Hz operation on the following Mac computers: … MacBook Pro (Retina, 13-inch, Early 2015) and later. MacBook Pro (Retina, 15-inch, Mid 2014) and later   Unfortunately it does not state whether two displays can be run at 3840 by 2160 at 60hz Stacks for Unity Launcher I'm looking forward to emulate the behavior of OS X Dock Stacks so that I can have, for instance, one stack for browsers and another one for IDES. People at ask ubuntu are doing a lot of cleaver stuff with custom Launchers and Quicklists (see ). So has anyone figured out how to have stacks in Unity? Anything fancier than a static Quicklist over a dummy folder icon? I have been thinking about ways of teaching electronics and I'm wondering if the following is true...  For starters, when we talk about voltage as energy per unit charge, is this energy manifest simply as the kinetic energy of the electron?  So, all other things being equal, will the electrons coming out of a 4 volt battery will have twice the velocity of electrons coming out of a 1 volt battery because of ${1 \over 2}mv^2$?  This would seem to imply that the voltage drop across different parts of the circuit is essentially a comparison of the electron speeds at those points.  Furthermore doesn't this imply that the electrons are nearly halted by the time they reach the positive terminal of the battery? This makes intuitive sense to me, it seems pedagogically sound, and it provides an explanation for how electrons "know" to give up their energy across the circuit -- electrostatic repulsion communicates later resistances to earlier parts of the circuit.  How much of the above is correct? The one things that makes me suspicious is that, in this picture, electron density at the positive terminal of the battery is very high, and repulsive forces would prohibit this. On the other hand, if the Joules per Coulomb are not coming from kinetic energy, then what? The field? But wouldn't that just dissipate as light?  Something is wrong but I don't know what. Can QGIS open KMZ files? I know I can add KML files once I have unzipped the KMZ files, but is there a way to open KMZ files directly?   When I try to open a kmz file on QGIS 1.8.0 on Windows XP, through the all files filter, it does not accept it as a valid format.   Am I doing something wrong?   Does this all depend on OGR support? It is discouraged to simply catch System.Exception. Instead, only the &quot;known&quot; exceptions should be caught. Now, this sometimes leads to unnecessary repetitive code, for example: try {     WebId = new Guid(queryString[&quot;web&quot;]); } catch (FormatException) {     WebId = Guid.Empty; } catch (OverflowException) {     WebId = Guid.Empty; }  I wonder: Is there a way to catch both exceptions and only call the WebId = Guid.Empty call once? The given example is rather simple, as it's only a . But imagine code where you modify an object multiple times, and if one of the manipulations fails expectedly, you want to &quot;reset&quot; the object. However, if there is an unexpected exception, I still want to throw that higher. Are security.ubuntu.com updates eventually merged into normal updates? Upon my observation, seems new packages will first appear in security.ubuntu.com, but if you upgrade later, they will appear in XX.archive.ubuntu.com (depends on your config),  Was that true ? Edit:  I foolishly changed my example right before posting this. The following is what I meant to ask.  On the wikipedia page below, it says "a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units."   Then, on the wikipedia page below, it says "an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials."   The page then gives an example of a polynomial that is irreducible "over the rationals" (I assume it is implied that "the integers" is referring to the ring of polynomials over the field of rationals, right?)  $p(x) = x^2 + 2 = (x+\sqrt{2})(x-\sqrt{2})$  On the Wofram page below, it says "the irreducible polynomials are examples of irreducible elements."   If irreducible polynomials are a subset of irreducible elements, how is $x+\sqrt{2}$ a unit?  What can it be multiplied by to equal the multiplicative identity?  I think I have it right this time (sorry. long day)  ====================================================================  The following is the original post (for those who already responded and those looking at the corresponding responses)  On the wikipedia page below, it says "a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units."   Then, on the wikipedia page below, it says "an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials."   The page then gives an example of a polynomial that is reducible "over the integers" (I assume it is implied that "the integers" is referring to some sort of ring, right? due to the claim below)  $p(x) = x^2 + 4x + 4 = (x+2)(x+2)$  On the Wofram page below, it says "the irreducible polynomials are examples of irreducible elements."   If irreducible polynomials are a subset of irreducible elements, how is $x+2$ a unit?  What can it be multiplied by to equal the multiplicative identity?