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Integral of $\sin x \cos x$ using two methods differs by a constant? $$ \int \sin \theta\cos \theta~d \theta= \int \frac {1} 2  \sin 2\theta~ d \theta=-\frac {1} 4  \cos 2\theta$$ But, if I let  $$ u=\sin \theta , \text{ then }du=\cos \theta~d\theta $$ Then $$ \int \sin \theta\cos \theta~d \theta= \int u ~ du =\frac { u^2 } 2 =\frac {1} 2 \sin^2 \theta $$ Since $$ \sin^2 \theta =\frac {1} 2 - \frac {1} 2 \cos 2\theta$$ The above can be written as $$ \int \sin \theta\cos \theta~d \theta= \frac {1} 2 \sin^2 \theta =\frac {1} 2 \left( \frac {1} 2 - \frac {1} 2 \cos 2\theta \right)=\frac {1} 4-\frac {1} 4 \cos 2\theta   $$ Why are the two results differ by the constant $1/4$? Thank you. Parse HTML in Android When I use ssh -X on my Mac (running OS X 10.6.7) to connect to my Ubuntu (11.04) box, I get the following warning:     Warning: untrusted X11 forwarding   setup failed: xauth key data not   generated Warning: No xauth data;   using fake authentication data for X11   forwarding.   Is there something I can do to make this warning go away?  If not, can I safely ignore it?  X11 forwarding seems to work fine, though I do see this message:     Xlib:  extension "RANDR" missing on   display "localhost:10.0".   Is that related to the warning?  (I'm guessing not.  If it's not, I'll file a new question about that.) What is the best way to include R code in the Latex paragraph I am a newbie in Latex. I have two simple questions regarding including R code in Latex.  Q1: For a block of R code, I use 'listings' package, as follows.  \documentclass[12pt]{report} \usepackage{listings} \begin{document} \begin{lstlisting}[language=R] &gt; dpareto(4, 2, 3) [1] 0.09375 &gt; dpareto(4, -2, 3) [1] NaN &gt; dpareto(seq(-1, 4, 1), 2, 3) [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.2962963 0.0937500 \end{lstlisting} \end{document}   I feel it's not professional. What is the best way to present the block of R code?  Q2: For code pieces in the paragraph, for example I wish to include a R function, or variable, or package name in my Latex context, what is the best way to do that? Finite abelian group contains an element with order equal to the lcm of the orders of its elements I will quote a question from my textbook, to prevent misinterpretation:     Let $G$ be a finite abelian group and let $m$ be the least common multiple of the orders of its elements. Prove that $G$ contains an element of order $m$.   I figured that, if $|G|=n$, then I should interpret the part with the least common multiple as $lcm(|x_1|,\dots,|x_n|)=m$, where $x_i\in G$ for $0\leq i\leq n$, thus, for all such $x_i$, $\exists a_i\in\mathbb{N}$ such that $m=|x_i|a_i$. I guess I should use the fact that $|x_i|$ divides $|G|$, so $\exists k\in \mathbb{N}$ such that $|G|=k|x_i|$ for all $x_i\in G$. I'm not really sure how to go from here, in particular how I should use the fact that $G$ is abelian. Prove that $f_n$ converges uniformly on $\mathbb{R}$ I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}}{\sum_{\mathbf{k}}1}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}}{\sum_{\mathbf{k}}1}\right)^2}\;.$$  Now, I want to do this for $d$-dimensions, so I convert this sum to integral.  $$L=\frac{\dfrac{\int_{0}^{k_f}k^{n} d^d k}{\int_{0}^{k_f}1d^d k}}{\left(\dfrac{\int_{0}^{k_f}k^{n-1}d^d k}{\int_{0}^{k_f}1d^d k}\right)^2}$$ where $d^dk  = \dfrac{d \pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1}dk$ is the volume element in $d$-dimensional space.  I want to get a simple expression for $L$ and how it depends with $k_f,n$ and $d$. any physical insight into this integral could also be helpful. Can you Dash twice on the same turn? The Dash action gives you extra movement for the current turn (equal to your speed). If I have an effect (like the Expeditious Retreat spell or the monk's Step of the Wind feature) which allows me to dash as a bonus action, can I also dash as my regular action, for triple movement? Book about a boy who travels to a parallel universe to get a book CASESAFEID() apex equivalent Is there any function in Apex which could emulate functionality of the  formula function? I'm trying to get 18-char Id variant from the standard one. Math Induction Proof: $(1+\frac1n)^n < n$ How do I know I have the correct exposure when shooting in manual mode? I have a Canon XTI which I've been using for a few years now. Most of the time I'm been shooting in one of the "semi-manual" modes like aperture priority or shutter priority, but now I'd like to try using full manual mode. I should have a new EF-S 17-55 f/2.8 delivered when I get home today. Before I was using a variable aperture Sigma 17-70 which made manual mode more difficult with the aperture changing arbitrarily.  I understand the effect of changing aperture and shutter speed, but right now I'm accustomed  to only controlling one at a time. The camera is usually setting the other to get the right exposure automatically for me. So if I'm controlling both how do I know I have the correct exposure? I understand I need to use the metering in my camera but I'm not sure exactly how do do that.   First when should I be using each of the metering modes (Evaluative, Partial, Center weighted, or Spot)? These just change the area of the scene which is metered correct?  How do I know what the meter should read off a specific object? Obviously if I just happen to be shooting a 18% gray wall then it should be at zero, but that can't apply to everything? If I'm shooting a black cat and take a spot meter reading off of the cat then the meter should read less then zero right? But how do I know how much less then zero? Are there rules of thumb I can follow? I've ordered a couple gray cards with my new lens. So experimenting with those should help.  How often should I be changing these settings? Say I'm at a party indoors which is consistently lit. I should be able to set the aperture, shutter speed, and ISO and just forget about it unless the lighting changes correct? Does the same apply to outdoor scenes? I've been hesitant to use manual mode in the past because I felt like I was spending a lot of times messing with settings before each shot. But not I'm realizing that this is really only the case when the lighting changes, or I want a different style of photo such as a different depth of field.  Finally, does exposure change with distance to the subject? For example say I want to take a picture of a statue with correct exposure using a gray card. I'd place the card on the statue, and while being careful not the block the light on the card find the correct exposure by spot metering the card. If I were then to step back 20' to get the whole statue in frame would I still have the correct exposure? 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. ecommerce item deleted by user, 301 redirect to HOME PAGE or 404 not found? I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world would go under long before an infinite amount of loops had been reached.  Are there any practical applications for the concept of infinity? Is it a useful concept in maths at all?  I know that Donald E. Knuth has argued that for all practical purposes, a very, very large number has the same effect as infinity, in his book "Things a Computer Scientist Rarely Talks About" (can't remember the exact quote, nor find it online, unfortunately).  Examples are appreciated. Institute Name in @masterthesis Citations I need to cite master thesis and the lecture notes into a document. But the problem with @masterthesis and @unpublished citations is they don't print an institution name in the pdflatex output of bibliography.   `\documentclass{article}   \usepackage{filecontents}  \usepackage[ backend=biber, style=numeric, sortlocale=de_DE, natbib=true, url=true,  doi=true, eprint=false ]{biblatex} \addbibresource{bibliography.bib}  \begin{filecontents}{bibliography.bib} @masterthesis{bhgt, title={Bike Model}, author={Fris}, year={2016}, school={University A}, address={XXX} } @unpublished{Map, title = {Electrical Engineering Book}, author={Koch}, howpublished = {Lecture Notes}, Institution= {University B}, year = {2015} } } \end{filecontents}  \begin{document}  Citation Test:  \cite{bhgt} and \cite{Map} \printbibliography  \end{document}`   I get this bibliography output:         [1] Fris. Bike Model. XXX, 2016.   [2] Koch. “Electrical Engineering Book”. Lecture Notes. 2015.       Any suggestions, how we can also print an institute name? Easy way to earn money on Pokemon Omega Ruby/Alpha Sapphire? I've recently been attempting to complete the Hoenn pokedex. I need a lot of pokéballs and ultra balls to catch em' all. If anyone knows an easy way to do this please simply explain it. I believe that repeatedly challenging the Elite Four may work, but I don't really want to do that. How do I pass parameter TO a Visualforce page from a link in an email? Here's what I am trying to do:  1) User creates record A.  2) Time-based workflow sends an email to the Record A creator based on a date field in the record.  3) The email template has a URL link.  4) The URL links to a Visualforce page that is basically a form with fields to be filled out (this Visualforce page is in a managed package) and a save button that creates a new User record.  5) When the User clicks the URL link in the email, the Visualforce page opens in a new window.  6) I would like to pass the values of the fields in record A to pre-fill the Visualforce page fields.   Here's my link:  &lt;a href="https://cs19.visual.force.com/apex/CloneThisUser?Desktop=true&amp;Id={!Pending_User__c.User_to_CloneId__c}&amp;formGroupInputSmall={!Pending_User__c.First_Name__c}   And here's the code from the relevant section of the Visualforce page:  &lt;div class="form-group form-group-sm"&gt;       &lt;!-- &lt;apex:outputLabel styleClass="col-sm-2 control-label" for="formGroupInputSmall"&gt;First Name&lt;/apex:outputLabel&gt; --&gt;       &lt;div class="col-xm-11 col-sm-12"&gt;           &lt;apex:inputField html-placeholder="First Name" styleClass="form-control FirstNameInput"  value="{!u.FirstName}" id="formGroupInputSmall"/&gt;       &lt;/div&gt;     &lt;/div&gt;   I have only included the first field I would like to pass the parameter to, in order to keep this post simple.   The link passes the User_to_Clone_Id__c just fine but does not fill in the "First Name" field with the value of the First_Name__c .  What am I doing incorrectly? Thanks. How to prepare files for rsync on a case insensitive filesystem? I am transferring a large number of files on a HFS+ filesystem.  The files are currently on ext2 partitions.  I have conflicts due to case insensitivity of the target partition (HFS+).  I want to identify the files that have duplicates filenames once they are in lower case, and delete them if they are actually duplicates.  I also found that I will have duplicate folder names if I convert everyhing to lower case. Basically these hard drives contain years of unsorted data, and I happen to have this problem with folder names too.   Does this seem reasonable:  find . -type f | while read f; do echo $f:l; done | sort | uniq -d    $f:l is ZSH for convert to lower case.  Now I want to keep only one instance of each file that have duplicates. How to do this efficiently ?  I do not want to find files with duplicate content, unless they have the same lower case filename. I will deal with duplicates later. I googled it but I couldn't find the answer... maybe I didn't seek it in the proper way. My question is if &quot;to&quot; must be repeated in sentences such as the following ones:   In the past, women used to do the housework and to take care of children.  The aim of this meeting is to control their progress and to discuss the new working plan.    If I had to bet, I would omit the second to... But I am not sure about it.