The triple integral:  $$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{(1+x^2+y^2+z^2)^2}=\frac{\pi^2}{32}$$ can be confirmed using Mathematica.   But how can one show it by hand? I provide two answers below: ,. Are there any other methods?    [Context added later by Jack:]   See also  calculation from Wolfram Alpha:            [Added later by OP:] The triple integral was fabricated by the method of the integral representation of  $1/p^2=\int_{0}^{\infty} t e^{pt} dt.$ See my first solution in the Answer below. The final result was confirmed numerically by Mathematica. More interestingly Mathematical gives this integral $I=\pi^2/12-A$ in analytic mode where A is Ahmed's Integral! which equals $5\pi^2/96$.
As in the title I have to evaluate this triple integral: $$\iiint_{[0,1]^3}\frac{dx\,dy\,dz}{(1+x^2+y^2+z^2)^2}$$ I've been trying to solve this since a week ago. The first thing I've done was understand the meaning of the integral. I think this integral represents the mass (as an example) of a unitary cube which contains materials of different density. The values of the materials density are, point for point, the inverse of the square of spheres centered in the origin plus one. The max value of the density is $1$ in the origin of the cube and the min value is $\frac{1}{16}$ on the opposite vertex. I suppose that the value of the integral is $\frac{\pi^2}{32}$ I've tried to use simple substitutions without any results, so I tried to change the coordinates with spherical and cylindrical systems. The spherical coordinates give me an incredibly long sum of integrals and I doubt that they're all integrable as elementary functions. The cylindrical gives me the following result $$\frac{\pi^2}{16}-\int_0^\frac{\sqrt2}{2}{\frac{\arctan{\sqrt{\frac{u^2-1}{u^2-2}}}}{\sqrt{2-u^2}}du},$$ which I'm not able to solve. My instinct tells me that there is a trick in some steps where I can observe that a difficult integral actually is exactly half of another one simpler but I can't figure out where. I'll appreciate any kind of suggestions.
stop Ipod from asking for old Apple ID password Recently, I've changed my Apple ID because my old E-mail has been hacked. Now that I set the new one as my apple ID, the old Apple account still appears when I open almost all my apps on my Ipod Touch (4th Gen.), asking for the password to my old Apple ID. When I put in the old password to my old Apple ID, it'll keep asking for it no matter how many times I type it in. How do I fix this while keeping my account secure? Any suggestions would help! How do I do a SELECT * INTO [temp table] FROM [stored procedure]? Not FROM [Table] and without defining [temp table]?  Select all data from BusinessLine into tmpBusLine works fine.  select * into tmpBusLine from BusinessLine   I am trying the same, but using a stored procedure that returns data, is not quite the same.  select * into tmpBusLine from exec getBusinessLineHistory '16 Mar 2009'   Output message:     Msg 156, Level 15, State 1, Line 2   Incorrect syntax near the keyword   'exec'.   I have read several examples of creating a temporary table with the same structure as the output stored procedure, which works fine, but it would be nice to not supply any columns. Setting CSS pseudo-class rules from JavaScript Automating the scanning of graphics files for corruption Does anyone know of a way to check graphics files (particularly JPEG, GIF, and PNG) for corruption (preferably in an automated way)?    Explanation:  A few days ago, a command worked incorrectly and ended up deleting thousands of graphics files from a FAT32 volume that was practically out of space. I’ve used several different file/photo-recovery programs, but naturally, they are limited in how much they can recover (though fortunately the volume has 8KB clusters, which helps somewhat).  Anyway, some of the larger files, that were fragmented, are now corrupt. Some of them are not even real files at all (the recovery software merely dumped the clusters that were pointed to by now-overwritten directory entries), while others are broken because of fragmentation.  Moreover, because some picture formats embed a smaller version of the picture as a thumbnail, scanning the thumbnails for corruption is not reliable because it may be intact while the actual file (i.e., the picture when viewed full-size), could be corrupt.    Here are a couple of examples:      Here’s the second one. It’s so damaged that it doesn’t display anything.    (A third one wouldn’t even upload because it doesn’t even have the correct header!) Why use "need not" instead of "do not need to"? The header of  states:     High-level languages need not be slower than low-level ones.     Why use need not instead of do not need? What does it mean? Also, why no to before be? Why there are real numbers with infinite digits, but no such natural numbers (or another reason why real numbers are uncountable) This question is me trying to understand (again) why there can be no one-to-one correspondence between the sets of natural and real numbers.  The source of confusion is this: if we abstract completely from the properties and operations on numbers and just think of them as strings of digits, it appears to be possible to list all the real numbers between $0$ and $1$ and all positive integers having the same digits in the same (or reverse) order, for example:  $$0.1 \to 1$$  $$0.01 \to 10$$  $$0.0101 \to 1010$$  $$0.110101 \to 101011$$  I use binary expansion for simplicity.  Since irrational numbers have infinite expansions, we would have to consider some kind of 'infinite natural numbers', like:  $$101110001\dots$$  or  $$\dots101110001$$  I think the main (and only) reason why such numbers don't exist is this requirement:   We need the sets of real and natural numbers to be ordered. We need to be able to do arithmetics on real and natural numbers.   Obviously, if we consider any of two types of 'infinite natural numbers' and the usual absolute value, we find that it's impossible to order them, or do arithmetics with them.  The only way is to change the norm (to $p-$adic norm for example).  While with the real numbers it's still possible to order them, and do arithmetics with them when they have infinite amount of digits in their expansion (or at least in principle).     Is my reasoning correct? Are the two requirements (ordering and arithmetics) enough to explain why natural numbers with infinite amount of digits do not exist, while real numbers do?   This confusion is not restricted to me alone, see  for example How can I get a DisplayLink USB monitor work under Ubuntu? I have a DisplayLink-based external USB monitor, which has both power and data over USB, and seems to work perfectly under Windows 7, but only can display a text console under Ubuntu 10.10, and that I can only use when I am actually switched to it. So the only Use I can have so far is to have some text-based monitoring or console that I can watch while working on the graphic display.   I know there are some development done for DisplayLink, but I never could get it to actually run properly.   Yes, I tried the detailed setup described in other , but they did not work, and instead crashed my X that I had to restore.   Merci :-) I flagged a post today, typed out a fairly thorough explanation, clicked the button... and got told that I'd already flagged this post (for something else, which I'd forgotten about) so this flag wasn't allowed.  (This happens if the first flag is still active.)  The dialogue with my explanation was gone at this point, so if I wanted to communicate with the moderators about that, I was going to have to start over.  (In case you're wondering how this could happen, one flag was related to the post itself and one was related to the user -- there being no way to flag a user directly, you have to hang it off of a post.  But I've also seen this with cases where the first flag hangs around unhandled for a week or two and I've forgotten about it.)  Since the system is going to detect that I've already flagged a post and reject a second flag, can it do that after I click the "flag" link instead of waiting until the end of the process?  (Or gray out the link so I can't click it in the first place?)  2015-12-18: someone suggested to me in chat today that this had been implemented, so I tested again.  The behavior has not changed: I flagged a post using "other", then flagged it again (also "other") before the first flag was handled, and the flag was rejected only after I'd typed the message.  (I did this on a site where I'm a moderator so I could then clean up after myself.) If a Matrix Has Only Zero as an Eigen-Value Then It Is Nilpotent I just installed the lastest version of . I works fine but in order to use it, I have to enter the following in the terminal: cd ugene-1.11.5/ then execute ./ugene -ui to launch UGENE GUI.   This takes too long. I would like to build a .desktop file for UGENE to be used in Ubuntu 12.04 so that I can simply open it from the unity launcher.  Not sure if this is helpful but the website says: By default ugene script launches the command-line version of UGENE.  Here's what I have (not sure if it's right):  [Desktop Entry] Version=1.0 Name=my Unipro UGENE GenericName=Integrated Bioinformatics Suite Comment=Unipro UGENE is a cross-platform visual environment for DNA and protein sequence analysis. MimeType=text/english;text/plain;text/x-makefile;text/x-c++hdr;text/x-c++src;text/x-chdr;text/x-csrc;text/x-java;text/x-moc;text/x-pascal;text/x-tcl;text/x-tex;application/x-shellscript;text/x-c;text/x-c++; Exec=/home/samuel/ugene-1.11.5 ./ugene -ui -c %F Icon=/usr/share/icons/hicolor/scalable/apps/ugene.svg Type=Application Terminal=true Categories=Utility;Development; StartupWMClass=UGENE   I added the launcher named my Unipro UGENE to my unity bar and tried opening it but it doesn't appear. I must be missing something. I don't know what the StartWMCClass field is suppose to be nor do I know whether I entered the correct MimeType. Better explanation of the common general relativity illustration (stretched sheet of fabric) A command-line clipboard copy and paste utility? In Windows I used command-line clipboard copy-and-paste utilities...      pclip.exe and gclip.exe     These were UnixUtils ports for Windows (but they only handled plain text). There were a couple of other native Windows utilities which could write/extract any format.    I've looked for something similar in Synaptic Package Manager, but I can't find anything.    Is there something there, that I've missed? ... or maybe this is available in Bash scripting?    The type of utility I'd like will be able to read/write via std-in/std-out or file-in/file-out, and handle Unicode, Rich Text Format, picture, etc. clipboard formats...    NB: I'm not after a clipboard manager. Intersection Number of $B = Y^2 - X^3 + X$ and $F = (X^2 + Y^2)^3 - 4X^2Y^2$ using the fact $I(P,F \cap B) = ord_P^B(F) $. hard voting, soft voting in ensemble based methods I'm reading . Then I'm not able to figure out the difference between hard voting and soft voting in context to ensemble based methods.  I quote descriptions of them from the book. The first two images from the top are description for hard voting, and last one is for soft voting.        In my view hard voting is majority decision, but I don't understand soft voting and the reason why soft voting is better than hard voting. Would anyone teach me these? Why does sudo change the PATH? Please help me  to evaluate combinatorially the following sum: $$\sum_{k=0}^n \binom{n}{k}$$   Thank you. How to drive multiple LEDs from a Li-ion battery (with varying voltage)? Application of Liouville's Theorem Let $f(z)$ be an entire function such that $$|f(z)|&lt;\frac{1}{|\text{Im}(z)|},\qquad z\in\Bbb C-\Bbb R.$$ The question asked me to prove that $f(z)=0$. At least looking at it, it really seems to have an application of Liouville's theorem lurking around somewhere, but I haven't found it.   My thoughts first led me to think about doing this by contradiction and using Picard's little theorem. So, I've considered a strip containing the real axis (say of width $2$ for simplicity). This will imply the complement maps into the unit disk, which implies that this strip has to map almost everywhere else on the complex plane. Now, on the imaginary axis, I know that $f(z)$ will vanish as $z\to\infty$, and this seems like it might be useful. I'm sort of under the impression $f(z)$ might have an essential singularity at $\infty$, which would mean $f(1/z)$ has an essential singularity at $0$. Either way, it of course can't have a pole at $\infty$ because of $f(z)$ vanishing on the imaginary axis, and if it is a removable singularity, it must be $0$, which still gives the solution. Therefore, I think it must have something to do with $f(z)$ having an essential singularity at $\infty$.   I was hoping to combine this with the above inequality and deduce a contradiction, but I haven't thought of one yet.   Any hints or suggestions? I'm studying for a prelim. exam, so this isn't homework.  Update: So, in the spirit of searching methods using the essential singularity at $z=0$, we have $$\left|f\left(\frac{1}{z}\right)\right|\leq\frac{1}{\left|\text{Im}\left(\frac{1}{z}\right)\right|}=\frac{|z|^2}{|y|}.$$ Thus, for $|y|&gt;1$, we have $$\left|f\left(\frac{1}{z}\right)\right|&lt;|y|\left|f\left(\frac{1}{z}\right)\right|&lt;|z|^2.$$ Thus, if I can show that $|f(1/z)|$ is a polynomial, we'll know it can't have an essential singularity at the origin, concluding my proof. SELECT name, gdp/population as capita  FROM world w1  WHERE continent = 'Europe'       AND capita &gt;  (     SELECT gdp/population      FROM world      WHERE name = 'United Kingdom' );   How come I can't refer to capita in the WHERE clause? It returns this error:  Unknown column 'capita' in 'where clause'   I can however, run the query if I replace capita with gdp/population.  I don't want to do gdp/population &gt; .... How can I do capita &gt; ...? Or is this not possible? (a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$  (b) $S = \bigcup_{k=1}^\infty S_k$   Work: For (a), I am not too sure about what approach I should use. I think finding a bijective function between the $S_k$ and $\mathbb{N}$ is hard to do . I also thought about using the Schroeder-Bernstein theorem and find injective functions $f:\mathbb{N}\rightarrow S_k$and $g:S_k\rightarrow\mathbb{N}$ in order to prove that $S_k$ and $\mathbb{N}$ are numerically equivalent. However, I have a hard time finding and proving these injective functions. I haven't really looked at (b) as I have yet to complete (a).