How to evaluate the following integral ? $$\boxed {\int_{0}^\infty \frac{\sin(t)}{t}\mathrm dt}  $$
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm  dx = \frac \pi 2$$ Well, can anyone prove this without using Residue theory?  I actually thought of using the series representation of $\sin x$: $$\int\limits_0^\infty \frac{\sin x} x \, dx = \lim\limits_{n \to \infty} \int\limits_0^n \frac{1}{t} \left( t - \frac{t^3}{3!} + \frac{t^5}{5!} + \cdots \right) \,\mathrm  dt$$ but I don't see how $\pi$ comes here, since we need the answer to be equal to $\dfrac{\pi}{2}$.
How to Disable Copy Paste (Browser) I am already dual booting windows 10 with Linux Mint, and I am trying to replace Mint with Ubuntu. I have already created a bootable usb for Ubuntu.  When I try to install Ubuntu, I don't see any option to install Ubuntu alongside windows 10. Rather, I see "install Ubuntu alongside Linux Mint". I didn't find any option to keep my windows 10. How can I replace Mint with Ubuntu in this dual boot system? Mint is installed on an ext4 partition and I have a Linux swap partition. What is a NullPointerException, and how do I fix it? What are Null Pointer Exceptions (java.lang.NullPointerException) and what causes them?  What methods/tools can be used to determine the cause so that you stop the exception from causing the program to terminate prematurely? Extend a language with additional keywords? Suppose a particle only influenced by gravity is going from point a to point b on an x,y plane. Is there an equation in which the particle takes the longest amount of time, yet still makes it to point b? I know the equation for brachistochrone is the parametric for an inverted cycloid, yet I'm not sure of the opposite. For reference, I am a high school student wanting to try this for a research project. I really only need to know whether this has already been done or not, but attempted solutions are obviously appreciated. Every time i publish a new post i want to display an icon like a badge (NEW) and after a specific time it will disappear.  Lets say i publish a new post and i want to display that it is a new post. So i want to insert an icon that for example a star, in each new post in the blog list view into the metadata of the posts. Also i want to be removed after a specific time will pass. Lets say in one day or 24 hours the icon should be removed automatically.  How can i do this? I really love Eclipse (PDT)... But i'm developing sites using HTML5 and CSS3 and it mark the HTML5 tags like &lt;nav&gt;, &lt;section&gt; and &lt;article&gt;  as invalid tags (inexistent).  How can I add those new tags to it?  And what about CSS3 rules?  I read something about remove the "invalid tag" check... and that's not what I want. :( Does a first generation iPod touch have the ability to download apps and use emojis? I am getting a 1st gen iPod touch for a friend and I was wondering if you can download like apps and games on it and if it has emojis? It would help to get an answer before I buy it! Are background jobs and foreground jobs always in running state? A job can be running or stopped (suspended).  When we are talking about foreground jobs and background jobs, are we only talking about running jobs?  Are background jobs and foreground jobs always in running state?  Does it make sense to say whether a stopped job is foreground or background? Using the expression "the same" for a previously mentioned item Why $SU(3)$ and not $U(3)$? Having different horizontally  distributed equations inside an align environment, only lines are numbered, but not individual equations.    How can the left equations be numbered using subequations? See also the question   \documentclass{article} \usepackage{amsmath} \begin{document}   \begin{subequations}     \begin{align}        %\label{a}         a &amp;= a ,&amp;       \label{b}         b &amp;= b ,\\       %\label{c}         c &amp;= c ,&amp;       \label{d}       d &amp;= d .     \end{align}   \end{subequations} \end{document}     The solution should be able to either:    attach a label to the left equations, or modify the labels on the right to read (1a,b) and (1c,d)    I couldn't find anything in the documentation of amsmath or other packages. How to keep vehicles upgrades equipped in Jetpack Joyride? I have Jetpack Joyride for Android, but every time I start the game I have to equip the magnetic upgrades to all vehicles, except the teleporter, which is the only one already equipped.  I had once a game crash that undid my last 2 magnets purchases, so I ended up with only the teleporter and 20k less in cash. After that, I've bought every vehicle magnet, but still the only one that will remain equipped is the teleport.  Anyone already found some similar bug? How could I fix this, it's kinda annoying have to mannually enable 5 different upgrades every time I want to play. I met an interesting issue about C#. I have code like below.  List&lt;Func&lt;int&gt;&gt; actions = new List&lt;Func&lt;int&gt;&gt;();  int variable = 0; while (variable &lt; 5) {     actions.Add(() =&gt; variable * 2);     ++ variable; }  foreach (var act in actions) {     Console.WriteLine(act.Invoke()); }   I expect it to output 0, 2, 4, 6, 8. However, it actually outputs five 10s.  It seems that it is due to all actions referring to one captured variable. As a result, when they get invoked, they all have same output.  Is there a way to work round this limit to have each action instance have its own captured variable? When I visit a site I haven't visited (while logged in) for a while, I see:  Welcome back! If you found this question useful, don't forget to vote both the question and the answers up.  That's often intended behavior. But today I visited  and was shown that banner even though the question and all its answers are locked. Voting arrows aren't even shown on any of the posts on that page. I believe that banner should be disabled on pages where there is no post that can be voted on by anyone. How to visualize Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$? pp  and ,   illustrates Triangle Inequality. But without referring to Triangle Inequality at all, how to picture  Reverse Triangle Inequality for all $\mathbf{a,b} \in C^n$? I don't want proofs or rigour. Firstly, I apologise if my thinking is completely incorrect - I am by no means a physicist (yet!), I have included layman illustrations to hopefully assist in explaining my thinking.  I am imagining the big bang to be, for lack of a better word, an explosion, thus the universe to be progressively expanding:    Therefore, to me it seems logical to suggest that there is an edge.  If this is the case, then is it reasonable to suggest that explosion + edge = 'bright edge'? Where light projects in all directions at the edge, not just outwards into the 'nothingness'.     So, this brings me to my question: why, when someone looks deeply enough 'at the edge' do they see an absence of light?   Again, apologies if I'm jumping to irrational conclusions &amp; am way off. My keyboard doesn't have an Fn key so I can't get the coordinates to show up. I don't know how to get into the multiplayer settings and Shift + F3 doesn't work either.  How can I fix this? Using relative paths for Windows shortcuts Does xbox live gold membership stack when I purchase a new one? I currently have an Xbox Live Gold membership that will expire in March 2011. However the price of membership is increasing to $59.99 starting November 1, 2010.   If I buy a 12 month membership now at the current price of $49.99, will the additional 12 months be added on to my existing membership or will I start a new membership which will expire 12 months after I buy it?