question: $\int_0^\infty\frac{\sin^2x}{x^2}dx$ is equal to (A)$\int_0^\infty\frac{\sin x}{x}dx$ (B)$\int_0^\infty\frac{\cos x}{x}dx$ (C)$\int_0^\infty\frac{\cos^2x}{x^2}dx$ (D)None of the above  I tried splitting the integral as $$\int_0^\infty\frac{1-\cos^2x}{x^2}dx=\int_0^\infty\frac{1}{x^2}dx-\int_0^\infty\frac{\cos^2x}{x^2}dx$$ But for the first integrand, the function is not defined when $x=0$. Then I tried using $$\sin^2x=\frac{1-\cos 2x}{2}$$ but that did not work as well.
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think?  It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm dx$$ is also $\frac{\pi}{2}.$ Many proofs of this latter one are already in this .
What does the tilde (~) mean at the end of a filename? What is the file with the ~ at the end of the filename for?  $ ls # aliased to add flags -rwxrwxr-x  1 durrantm  2741 May 16 09:28 strip_out_rspec_prep_cmds.sh~* drwxrwxr-x 13 durrantm  4096 May 16 14:21 ../ -rwxrwxr-x  1 durrantm  2221 May 16 14:58 strip_out_rspec_prep_cmds.sh*   This is not the same as .swp files which are there while editing.  The two files have quite a few differences and the newer file (no ~ at the end) has the most recent changes and those changes are not in the older (~) file.  Looks like I can delete it? Is there an alternative to DVD-Shrink, that can compress a DVD image to a smaller size to fit on a writeable DVD? Is there a linux alternative to the Windows application DVD-Shrink, that can compress a DVD image to a smaller size to fit on a writeable DVD?  In backing up some DVD's the ripped ISO file size is over 7000MB is there an alternative linux software, similar to Windows "DVD Shrink" that can compress this to fit on a writeable DVD which is about 4000MB I know in TNG that all computer interfaces are now touch-screen, and that got me thinking: when was the first reference to touch-screen computer interfaces in Sci-Fi made? HTML Element Array, name="something[]" or name="something"? How to echo a bang! Importing google-play-services lib into Intellij IDEA 12 (and 13) In doing instanton problems or when connecting quantum field theory to statistical mechanics, I often see people trying the Wick rotation trick by defining an imaginary time $\tau\equiv it$. So, in this case, we should make sure that $\tau$ is imaginary, right? However, I often see places where people seem to handle $\tau$ as a real number, for example setting $\tau$ to $\pm\infty$. Moreover, I remember in one of my quantum mechanics classes, the professor said the Euclidean time $\tau$ was real. I am so confused and need some good explanations. Macbook Air - how to install a dual bootable Ubuntu 14.04 LTS On my Macbook Air 10.9.2 without crashing the current setup, how can i install as dual boot: Ubuntu 14.04 LTS ? (can anyone please advise some step by step please) Combinatorics Question Asking How Many Integer Solutions Given Different Restrictions How many integer solutions of $x_1 + x_2 + x_3 + x_4 = 28$ are there with:  (a) $0 ≤ x_i ≤ 12$?  (b) $−10 ≤ x_i ≤ 20$?  (c) $0 ≤ x_i, x_1 ≤ 6, x_2 ≤ 10, x_3 ≤ 15, x_4 ≤ 21$?  \ \  My attempts (I was really only able to solve part (a) but I would appreciate if you could check my work for this one as well):  (a) First I took the total number of integer solutions with no restrictions: There would be ${28+3\choose 3}$ = ${31\choose 3}$ total solutions with no restrictions.   Then I used the inclusion exclusion principle and knew I would have to subtract [P(A) + P(B) + P(C) + P(D)] where A --> $x_1$ is greater than or equal to 13, ..., D--> $x_4$ is greater than or equal to 13. The probabilities of all these cases would equal ${15+3\choose 3}$ = ${18\choose 3}$ because once we assign 13 to one of the $x_i$ there would be $28 - 13 = 15$ remaining to distribute among the $x_i$ values. We would also need to choose which of the $x_i$ to give the 13 to so we would multiply the ${18\choose 3}$ by ${4\choose 1}$ = ${4\choose 1}{18\choose 3}$  We would then have to add back, since we "oversubtracted", the case in which there is an intersection between two of the $x_i$ or in other words two of the $x_i$ receive 13 or more. P$(A\bigcap B)$ = $28-2(13) = 2$ remaining to distribute among the $x_i$ values. So we would also have to choose which of the $x_i$ to give the 13 to so we would have ${4\choose 2}$ ways to choose them. So there would be ${4\choose 2}{2+3\choose 3}$ = ${4\choose 2}{5\choose 3}$ solutions in this case.   Thus the number of solutions would be: ${31\choose 3} - {4\choose 1}{18\choose 3} + {4\choose 2}{5\choose 3}$.  I'm not sure if this is correct and I'm really stuck on b and c so I would appreciate any help or feedback. Thank you! Why there's a finite $\Gamma '$ such that $\Gamma ' \vDash \varphi$? Suppose $\Gamma$ is an infinite set of formulas in FOL. It is given that for a structure, $M$: If $\Gamma$ is satisfiable in $M$ then $\varphi$ is valid in $M$.  Why is it true that there's some finite $\Gamma ' \subset \Gamma$, such that $\Gamma ' \vDash \varphi$?   It reminds the compactness theorem in a way, but it's not quite the same. Were there any canon examples of Harry being an innately powerful wizard? Was Harry Potter an innately powerful/skillful Wizard in any way, based on canon?  It seems that whenever Harry Potter performed some unusually potent bit of magic (potent as far as require raw power AND/OR skill), most of the time we find that it wasn’t innately Harry’s skill, but a combination of:   some extraneous magic forces/rules/powers circumstance great reaction time (which is obviously extraordinary from his Quidditch playing; and is likely a larger factor in his dueling than the strength of his magic) general smartness or (the part that Harry’s detractors, including Voldemort missed) Harry’s strength of character/personality   The only two counter-examples I can think of which show him as having extraordinary or at least obviously above average magical abilities are:   His skill at flying brooms - while a part of it is pure physical skill, at least part is magical. His (corporeal) Patronus charm   As far as examples of his unusual feats being due to reasons listed above:   Obviously, surviving Avada Kedavra as an infant - due to Lily’s sacrifice Parselmouth talent - proxied via Voldemort’s piece of soul. Ability to see the Sorcerer’s Stone in the Mirror of Erised - due to purity of his character, nothing magical. Obvious lack of talent at potion making throughout all of the books (sans cheating in HBP) His win over the Basilisk - due to Hermione’s research, character traits, and as a consequence of the last one, getting Fawkes and Gryffindor’s Sword’s help.  Just to be clear, before someone flames me - what he did was obviously super extraordinary, BUT IMHO he would have achieved the same result if he was turned into a Squib for the duration of Chamber of Secrets fight. TriWizard Tournament entry - wasn’t him, it was Barty Crouch TriWizard Tournament second challenge - a combination of character and outside help (someone told him about Gillyweed) TriWizard Tournament third challenge - Mostly, Barty Crouch’s cheating  Note that I’m omitting the first challenge, since that one was the aforementioned Broom flying skills. His survival in the cemetery after Voldemort’s return - again, a combination of superior strength of character and super powerful wandlore magic that had nothing to do with his skill as a Wizard. The whole “last Horcrux” thing when surviving the second Avada Kedavra from Voldemort His ownership of the Elder Wand in the very last duel with Voldemort.   So, to paraphrase my original question, was there any canon/JKR evidence that Harry was anything but a completely average/mediocre wizard in any way, shape, or form if you take into account only his own magical abilities and skills? I've tried to import but I don't see the (.max) file as an option when importing. Is there another way to import the 3DMax model into Blender for rigging and animation? Killing thread after some specified time limit in Java I (think) that I've configured a static IP address in /etc/network/interfaces:  # The primary network interface  auto eth0  iface eth0 inet static     address 10.1.1.2     netmask 255.255.255.0     network 10.1.1.0     broadcast 10.1.1.255     gateway 10.1.1.1   And I change the file /etc/resolv.conf to include the desired nameservers.   Then I restart networking sudo /etc/init.d/networking restart  But after some hours resolv.conf always reverts back to the DHCP nameserver.   I should also say that this is Ubuntu Server with only l0 and eth0 interfaces. If $[G' : G'']\leq p^2$, then $G'$ is abelian. Problem : Let $G$ be a p-group and $G'$ denote the commutator subgroup of $G$. If $[G' : G'']\leq p^2$, then $G'$ is abelian.  It is easy to prove it for the case of $[G' : G'']=1$ since G is solvable. But it is too hard to prove the other cases. Please give me any advice or solutions. Thanks in advance. How do I remove repeated elements from ArrayList? I have an ArrayList&lt;String&gt;, and I want to remove repeated strings from it. How can I do this? Interscholastic Mathematics League Senior B Division #2 Brand new disk brakes are barely stopping the bike C program to convert Fahrenheit to Celsius always prints zero I'm writing a bash script that needs to delete old files.  It's currently implemented using :  find $LOCATION -name $REQUIRED_FILES -type f -mtime +1 -delete   This will delete of the files older than 1 day.  However, what if I need a finer resolution that 1 day, say like 6 hours old? Is there a nice clean way to do it, like there is using find and -mtime?