The original quesion is $$\int_0^\infty\left(\frac{\sin x}{x^3}-\frac1{x^2}\right)\,dx$$ Can I divide them into two parts? Then using residue theorem?
Show that $$\int_{0}^{\infty} \frac{x - \sin(x)}{x^3} \, dx = \frac{\pi}{4}$$   My attempt is as follows: Let $$f(z) = \frac{z - i e^{iz}}{z^3}$$ and consider the contour on $[\epsilon, R]$ followed by a semicircular arc in the counter clockwise direction, then on $[-R, -\epsilon]$, then the semicircular clockwise contour avoiding the origin. We have, then, that  $$0 = \int_{\Gamma} f(z) dz = \int_{[\epsilon, R]} f(t) dt + \int_{C_R}f(Re^{it})Rie^{it}dt + \int_{[-R, -\epsilon]}f(t)dt + \int_{C_{\epsilon}}f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  Then the first and third integrals ( $I_1$ and $I_3$) combine so that  $$I_1 + I_3 = 2\int_{\epsilon}^R \frac{t - \sin{t}}{t^3}\,dt$$  Further,   $$|I_{C_R}| \leq \int_0^\pi \left|\frac{Re^{it} - ie^{-R\sin{t}}e^{iRcos{t}}}{R^2 e^{2it}} \right|dt \rightarrow 0 \text{ as } R\rightarrow \infty$$  (I've omitted the details, it isn't too bad to bound)  However, I'm having trouble computing the limit  $$\lim_{\epsilon \rightarrow 0}\int_{C_{\epsilon}} f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  No matter which way I look at it, it seems like this limit does not exist. Perhaps I'm seeing something wrong or have I chosen a bad $f(z)$?
Where is Jon going? At the end of S08E06 of Game of Thrones, Jon Snow along with Tormund and others of the Night's Watch      leave the Night's Watch.   and go into the woods.  Where are they going? And why? incoming mail just sits in the drop folder I want a document library to recieve email and store it so followed the steps outlined here:   I seem to be a bit stuck.  The mail's being delivered to the machine and it's appearing as .eml files in the drop folder, so SMTP and delivery is OK.    I assume Sharepoint's supposed to pick up the file, stick it in a library and delete it, but it doesn't, the mail file just sits there.  Nothing in the logs, obviously.  I haven't set this up before, how often does the timer service come about and pick up the mail? Equals(=) vs. LIKE When using SQL, are there any benefits of using = in a WHERE clause instead of LIKE?  Without any special operators, LIKE and = are the same, right? request: bragging & boasting but all made up of lies This is a single word, idiom, phrase, expression request - so basically I'll take anything that fits the bill.  Basically I'm trying to translate the Chinese word 吹牛 - which dictionaries will tell you mean to brag &amp; to boast - but it actually also refers to things that are usually all untruths and lies.  I thought, originally, that one of these would work: he's just bragging (i.e.: it's not, actually, true) - but Google only says:     brag      braɡ/      verb      ?verb: brag; 3rd person present: brags; past tense: bragged; past participle: bragged; gerund or present participle: bragging         say something in a boastful manner.   "he bragged that he was sure of victory"         noun      noun: brag; plural noun: brags         a boastful statement.      and     boast1      bəʊst/      verb      verb: boast; 3rd person present: boasts; past tense: boasted; past participle: boasted; gerund or present participle: boasting         talk with excessive pride and self-satisfaction about one's achievements, possessions, or abilities.      "she boasted about her many conquests"      synonyms:  brag, crow, swagger, swank, gloat, show off, blow one's own trumpet, sing one's own praises, congratulate oneself, pat oneself on the back; More      exaggerate, overstate;   preen oneself, give oneself airs;   informaltalk big, blow hard, lay it on thick, shoot one's mouth off;   informalskite, big-note oneself   "his mother had been boasting about how wonderful he was to all her friends"   antonyms:   deprecate, belittle   (of a person, place, or thing) possess (a feature that is a source of pride).   "the hotel boasts high standards of comfort"         noun      noun: boast; plural noun: boasts         an act of talking with excessive pride and self-satisfaction.      "I said I would win and it wasn't an idle boast"      Not really seeing anything talking about lies.   Anything about bragging or boasting that involves lies? Why does extended piston become stuck if air above it is powered? To reproduce this problem make layout like this and then power the piston. Once the piston is extended, removing the power source will not retract it:    Since piston head block is transparent, I see no explanation for this. Also, even if it wasn't transparent, it still wouldn't power a piston. For example this doesn't power the piston:    But if I place redstone and THEN place the piston, it's powered:    It also remains powered no matter what, as long as there isn't a block update event:    I suppose this behavior can be used to make a block update detector, but in my case this sucks. I wanted such setup for automatic smooth stone generator (without any clocks):    But due to this bug, it won't work - the piston will be always extended. I have previously saved a copy of /var/lib/mysql/ddms directory ("ddms" is the schema name). Now I installed a new MySQL on a freshly installed Ubuntu 10.04.3 LTS by running apt-get install mysql-server, I believe version 5.1 was installed. After I copy the ddms directory under /var/lib/mysql, some of its tables work fine, these are the tables with an associated set of three files: a .frm file, a .MYD file and a .MYI file.  However, there are two tables with a different set of files: a .frm file and a .ibd file. These two tables didn't show up in the table list in phpMyAdmin. When I look at the error log, it says:  [ERROR] Cannot find or open table ddms/dictionary_item from the internal data dictionary of InnoDB though the .frm file for the table exists. Maybe you have deleted and recreated InnoDB data files but have forgotten to delete the corresponding .frm files of InnoDB tables, or you have moved .frm files to another database? or, the table contains indexes that this version of the engine doesn't support.   Please help with restoring these two tables. Thanks. Random place change in row with restrictions Unresolved external symbol on static class members Very simply put:  I have a class that consists mostly of static public members, so I can group similar functions together that still have to be called from other classes/functions.  Anyway, I have defined two static unsigned char variables in my class public scope, when I try to modify these values in the same class' constructor, I am getting an "unresolved external symbol" error at compilation.  class test  { public:     static unsigned char X;     static unsigned char Y;      ...      test(); };  test::test()  {     X = 1;     Y = 2; }   I'm new to C++ so go easy on me. Why can't I do this? When does $\lim_{n\rightarrow \infty} a_n = 0$ imply $\sum_{n = 0}^{\infty} a_n$ converges? This is just a general question. I know the opposite statement is always true. I'm just asking when the converse holds. I know that the converse is not true in some cases, such as for harmonic series. What additional restrictions are needed to make this statement true? Why multiple capacitors in parallel? Wondering why you would choose to use smaller capacitors in parallel vs one bigger one.  Here's an example:      Is this a component size and track width choice?  Thanks! I'm trying to find the derivative of $\sin^{-1}(x)$. I know the steps that lead to $\frac{1}{\sqrt{1-x^2}}$, however I don't understand why the following reasoning leads to a wrong answer.  Because  $$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} $$  If we plug in for $f(x) = \sin(x)$, and because $\frac{d}{dx}\sin(x) = \cos(x)$ we get  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\sin^{-1}(x))} $$  Since $\sin(x) = \cos(x-\frac{\pi}{2})$, we can state that $\sin^{-1}(x) = \cos^{-1}(x-\frac{\pi}{2})$. (I suspect this is what is wrong)  Thus,  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\cos^{-1}(x-\frac{\pi}{2}))}$$  Then, by the definition of inverse function, we have  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{x - \frac{\pi}{2}}$$ This is a  about securing public DNS resolvers   Open DNS servers seem pretty neat and convenient, as they provide IP addresses that we can use consistently across our company regardless of where they are located. Google and OpenDNS provide this functionality, but I'm not sure that I want these companies to have access to our DNS queries.  I want to set up something like this for use by our company, but I hear a lot about this being dangerous practice (particularly in regards to ) and I want to make sure that we do this right. What things do I need to keep in mind when building this type of environment? Pre & post increment operator behavior in C, C++, Java, & C# Let $F$ be a field. Why does $F[x]$ have infinitely many irreducible elements?   For the case F has characteristic 0  Then x-a is irreducible for all a $\in F$ since x satisfies no non-trivial relations in F.  Obviously this argument fails for a finite field since there are only finitely many a to choose from.  So how may I construct irreducible polynomials in a finite field?  I figure it must involve higher powers of x, maybe $x^n-a$ ? How do I check if an object has a specific property in JavaScript? For $u:\mathbb{R}\times [0,1]$ with boundary conditions $u(0,x)=\cos (2\pi x)$ and $u(t,0)=u(t,1)$.   Solve $u_t=u_{xx}+u_x$.  I had this on an exam and tried to write $u$ as a product of two single variate functions and convert to an ode using the usual methods but things got messy and I couldn't finish. I also tried to write $u$ as a Fourier series in $x$ with coefficients $a_k(t)$ depending on $t$, but this also didn't seem to yield anything.  Is there a more clever way to approach this, possibly with Fourier series? I heard from a friend there may be a Fourier transform method since the equation was "homogeneous in momentum space."   Thanks! Catch Regex too complicated when splitting a string in salesforce How to customize title page in beamer so that it looks like How can I modify title page so that it looks like it is in  video. Please seek to 16 seconds and watch title page. I do not have permission to post images. I liked this theme and I want to make a beamer theme that looks similar. I tried with beamercolorbox but I failed to position the box to the bottom.      Edit: Here is my minimum working example  \documentclass[xcolor=dvipsnames]{beamer} \defbeamertemplate*{title page}{customized}[1][] { \usebeamerfont{subtitle} \usebeamercolor[fg]{subtitle}  {\flushright \setbeamercolor{author}{bg=white,fg=Red} \usebeamerfont{author} \insertauthor\par \usebeamerfont{institute}\insertinstitute\par }  \vspace{0.2in}  \setbeamercolor{postit}{fg=white,bg=YellowOrange}  \begin{beamercolorbox}[sep=1em,wd=1.062\textwidth,ht=3cm,dp=3cm,right]{postit} \usebeamerfont{title} {\huge \inserttitle}\par \usebeamerfont{subtitle} {\insertsubtitle} \end{beamercolorbox} } \usecolortheme[named=YellowOrange]{structure} \useoutertheme{infolines} \setbeamertemplate{navigation symbols}{}  \institute{Harvard College Professor and\\ Johnstone Family Professor in Psychology, Harvard University}  \title{SAY WHAT?} \subtitle{Linguistics as a Window to\\ Understanding the Brain} \author{Steven Pinker}  \begin{document} { \setbeamertemplate{footline}{} \begin{frame} \titlepage \end{frame} } \end{document} Events triggered by dynamically generated element are not captured by event handler I am trying to model a violin belly and I am intrigued, how is it possible to make faces with soft / rounded edges?    The edges I do are totally straight (see the blue amoeba).  Also, do you have some idea on how to avoid the edge loops getting too "compressed" in the middle of the model (see the red amoeba) ?    I know the Subdivision surface modifier, but I want curved edges after applying the modifier... like in this picture :  Any help will be very appreciated. Thank you in advance!  Marcos  2nd part  In fact, I've tryied to model using nurbs and bezier curves, but I wasn't able to find any tool similar to loft that works with curves, then I was forced to convert it in meshes, the results were not terrific and I've started to move the vertices manually, a real pain... Unlike a guitar, each edge loop is on a diferente elevation on z axis and I have to keep the original dimensions, So I have to be extremely careful when using the subdivision suface modifier. I've already modeled this think on three different CAD softwares, but the meshes are far from good... I want to improve the edge loops, but it will be a long path... Thank you again.