$$\int_0^\infty \frac{\ln x}{x^2+2x+2}dx$$ A question similar to this one has been asked before , but there the quadratic doesn't have a constant term.  I tried to use completing the square and use the answers in the linked question but since the log term in the numerator has $x$ not $x+1 $(after completing the square the quadratic is $(x+1)^2 + 1$) so that doesn't seem to work.
Evaluate  $$\int_0^\infty \frac{\log x \; dx}{x^{2} + 2x + 2}$$  by integrating a branch of $(\log z)^{2}/(z^{2} + 2z +2)$ along a keyhole contour.  The thing I have trouble with is why I should be examining the square of the log - I guess it has something to do with ln x in fact NOT being the real part of log z, since log z is (or can be) defined on the entire negative real axis and ln x can't. But looking at the square would definitely not have been my first plan of attack :(
How to delete all music on my iPhone How do I delete all the music on my iPhone.  I tried doing it through iTunes but it says my collection is not managed by iTunes and I don't want to erase and sync because all my apps are nicely backed up and sorted with iTunes.  I have only seen a way of deleting song by song on the phone. I have this, in D1:  =sum(D3:D)   I want D1 to always be the sum of the rest of column D (D2 is a header)  I sometimes move a row, from eg 8 to 3, and that causes D1 to change to sum(d4:d).    I want D1 to stay as my sum of the column. Is infinity a real number? Is infinity a real number?  If not, why not?  I want some very good arguments.   Thanks.  $$\rightarrow\leftarrow\Huge\Huge\Huge\boldsymbol\infty$$ Mathematics for Quantum Mechanics What math should I study if I want to get a basic understanding of quantum mechanics and especially to be able to use the Schrodinger's equation. Probability that no two consecutive heads occur? A fair coin is tossed $10$ times. What is the probability that no two   consecutive tosses are heads?   Possibilities are (dont mind the number of terms):  $H TTTTTTH$, $HTHTHTHTHTHTHT$.  But except for those,  let $y(n)$ be the number of sequences that start with $T$  $T _$, there are two options, $T$ and $H$ so,  $y(n) = y(n - 1) + x(n-1) = y(n - 1) + y(n - 2)$  Let $x(n)$ be the number of sequences that start with $H$,   $H _$ the next option is only $T$ hence,  $x(n) = y(n - 1)$   The total number of sequences $F(n)$ is:  $$F(n) = y(n) + x(n) = 2y(n - 1) + y(n - 2)$$  We are after $F(10)$,  $$F(10) = 2y(9) + y(8)$$  $$F(3) = 2y(2) + y(1) = 1 + 4 = 5$$  $$F(4) = 2y(3) + y(2) = 6 + 2 = 8$$  But I'm not quite sure where this will take me though. So I tried a few different answers that I found here, and I feel that this is possibly a duplicate question, but everything I have tried from the other solutions hasn't worked.  Basically I am trying to bake a material to a texture that will be used in Unity, but every test gives me a black background image.  In the image below, the top image texture is the mask (black and white, white being the leaves), and the bottom image texture is the colour. Why did Superman & Supergirl start to suffocate instantly when Luthor made our yellow Sun red? Spoiler Alert: If you haven't read all comics series &amp; story arc of New Krypton Saga viz. New Krypton, World of New Krypton, Last Stand of New Krypton, Superman: War of Supermen etc, drop this question right away!  While battling the evil alien Brainiac, Superman &amp; Supergirl discovered the long-lost bottled city of Kandor which was stolen from Krypton prior to its destruction. Superman re-enlarged the city on Earth, but it wasn't welcomed by Humans. Humans assaulted the city in which Kryptonian's leader Zor-El (Supergirl's father) was killed. Using highly evolved science, Alura (Zor-El's wife) created a new planet "New Krypton". Plus, she, as a new leader, freed General Zod from Phantom Zone to command new super military of New Krypton (every kryptonians had super powers due to yellow sun).  While New Krypton &amp; Earth were at cold war, Zod ordered his military to attack Earth. When military was in space, Sam Lane (Lois Lane's father) destroyed New Krypton by a trick making their race extinct again. Kryptonian military successfully crushed Earth's Mars defense facility &amp; they headed towards Earth.  When they were about to enter atmosphere of Earth:      Many Kryptonians died under the red sun because they couldn't survive in empty space. Its somewhat plausible because they weren't under yellow sun for long. But, what about Superman &amp; Supergirl? They did survive because they were under yellow sun for very long. But, why did they become very weak instantly? If the power draining rate by radiation of red sun was that high, how did Superman &amp; Supergirl survive?  Is this a plot hole? Or, am I missing something? Has anyone had any success using twitters typeahead.js within visualforce pages? I am trying to get  to work within salesforce visualforce pages.  I have a basic vf page from  example, but it does not respopnd, I suspect there may be some "unknown" workaround needed? Does anyone know?  &lt;apex:page standardController="Contact" showHeader="false" sidebar="false" docType="html-5.0"  standardStylesheets="false"&gt;  &lt;script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"&gt;&lt;/script&gt; &lt;script src="http://twitter.github.io/typeahead.js/releases/latest/typeahead.bundle.js"&gt;&lt;/script&gt;      &lt;div id="the-basics"&gt;   &lt;input class="typeahead" type="text" placeholder="States of USA"/&gt; &lt;/div&gt;  &lt;script&gt;     var substringMatcher = function(strs) {       return function findMatches(q, cb) {         var matches, substrRegex;          // an array that will be populated with substring matches         matches = [];          // regex used to determine if a string contains the substring `q`         substrRegex = new RegExp(q, 'i');          // iterate through the pool of strings and for any string that         // contains the substring `q`, add it to the `matches` array         $.each(strs, function(i, str) {           if (substrRegex.test(str)) {             // the typeahead jQuery plugin expects suggestions to a             // JavaScript object, refer to typeahead docs for more info             matches.push({ value: str });           }         });          cb(matches);       };     };      var states = ['Alabama', 'Alaska', 'Arizona', 'Arkansas', 'California',       'Colorado', 'Connecticut', 'Delaware', 'Florida', 'Georgia', 'Hawaii',       'Idaho', 'Illinois', 'Indiana', 'Iowa', 'Kansas', 'Kentucky', 'Louisiana',       'Maine', 'Maryland', 'Massachusetts', 'Michigan', 'Minnesota',       'Mississippi', 'Missouri', 'Montana', 'Nebraska', 'Nevada', 'New Hampshire',       'New Jersey', 'New Mexico', 'New York', 'North Carolina', 'North Dakota',       'Ohio', 'Oklahoma', 'Oregon', 'Pennsylvania', 'Rhode Island',       'South Carolina', 'South Dakota', 'Tennessee', 'Texas', 'Utah', 'Vermont',       'Virginia', 'Washington', 'West Virginia', 'Wisconsin', 'Wyoming'     ];      $('#the-basics .typeahead').typeahead({       hint: true,       highlight: true,       minLength: 1     },     {       name: 'states',       displayKey: 'value',       source: substringMatcher(states)     }); &lt;/script&gt; Is RSA key size the size of private key exponent? I have implemented a key pair generation scheme for RSA algorithm. I have taken the length of private key exponent as RSA key size, but then I've got to know that RSA key size is the size of the modulus $n$. Is it wrong to take RSA key size as the size of the private key exponent?  To generate a private key of 1024 bits, I increase the size of private key exponent regardless the modulus $n$. Since the final Black Hole (BH) had 3 solar masses less of mass than the original binary BH, it seems the 2 BHs lost mass, and with it event surface area, entropy, and information. If that came from inside the BHs it seems that it would be a contradiction of what a BH is --'nothing escapes its event horizon'. Some may say that came from the rotational energy of the pair, but the new BH now seems to have less than the two original ones summed. So, did some of that escape the horizons? Do the automatic success and failure rules apply to skill checks? We had a rule check tonight that we couldn't answer. On the Pathfinder wiki I could only find info on natural 20s and 1s in regards to attack rolls and saves in combat.  Do they also apply to skill checks? How to select records with maximum values in two columns? How can I prepend a tag to the beginning of several files? Find $\space\ \begin{align*} \lim_ {x \to+\infty} \left [ \frac{\log_{2}(x-1)}{x}\right]   \end{align*}$.   After some minutes around this limit I did it this way:  $\log_{2}(x-1)=y \Leftrightarrow 2^y=x-1$  So,$\space x=2^y+1$.  When $x \to +\infty$,$\space y \to +\infty$ also. By substitution:  $\begin{align*} \lim_ {y \to+\infty} \left [ \frac{\log_{2}(2^y+1-1)}{2^y+1}\right]=\lim_ {y \to+\infty} \left [ \frac{\log_{2}(2^y)}{2^y+1}\right]=\end{align*}$  $\begin{align*}\lim_ {y \to+\infty} \left [ \frac{y}{2^y+1}\right]=\lim_ {y \to+\infty} \left [ \frac{1}{\frac{2^y+1}{y}}  \right]=\lim_ {y \to+\infty} \left [ \frac{1}{\frac{2^y}{y}+\frac{1}{y}}\right]= \frac{1}{+\infty+0}=0  \end{align*}$  Is this correct?Are there any other easy way to find this limit?Thanks What security systems and processes are in place to prevent malicious third parties from hacking/compromising the security of code in the Debian mirrors, or to verify that the packages we get are in fact the ones the maintainers think they are? How do I assign a value to a whole array of integers? I really like the new sidebar. With the new top bar eliminating the &quot;Hot Questions&quot; area there, I think this is the perfect place for it. One-click access to time-wasting questions about topics I may or may not have an interest in. Seriously, it's a procrastinator's dream come true. The only suggestion I can add is to extend it. I assume the &quot;hot&quot; algorithm can spit out more than 12 questions, and since we now have a huge amount of room for it(without being restricted to a weird drop-down scrollbox), why not show more? If there's something else planned for that spot, disregard. At a minimum, the title could link to the , as ShaWizDowArd suggested in comments. I've been using SE for a couple years now, and I've never seen that page, so it would add visibility there.    Update: They added a link! Just noticed this, I'm pretty sure it wasn't there yesterday. When you click it, it loads 12 more. At the moment, the link disappears once clicked, so you can see a total of 24 here now. Not a link to the full list, but better: How do I join two separate objects so I can edit them as one mesh? Suppose $f:X\rightarrow Y$ is a surjection and you want to show that there exists $g:Y\rightarrow X$ s.t. $f\circ g=\mathrm{id}_Y$. You need the AC to show this.  However, suppose $f$ is a injection and you want show there is $g$ s.t. $g\circ f=\mathrm{id}_X$. Then, according to my textbook, you don't need the AC to show this.  This is counterintuitive to me, because it's like you need a special axiom to claim that an infinite product of big sets is nonempty, while you don't need one to claim that an infinite product of singleton sets is nonempty, which seems smaller than the former.  So why don't you need the AC to show the latter?  EDIT: $X$ should be nonempty.  EDIT 2: I realized (after asking this) that my question mostly concerns whether the AC is needed to say that an infinite product of finite sets is nonempty, and why. What is the need of acceleration? As we can usually predict the various forces that a body might be undergoing based on its position and its velocity.Hence what is the need of acceleration?