For Example:  $\int x^2 dx$  This is a very simple integral yet I have trouble grasping the meaning of various notations. Does the placement of '$dx$' next to $x^2$ mean multiplying $dx$ by the object of integration which is $x^2$?  If it did, then this would be solved by taking the integral of $x^2$ (this equals $\frac{x^3}{3}$) and multiplying it by the derivative of $x^2$ , the derivative of $x^2$ is $2x$.   $\frac{x^3}{3}$ $*$ $2x$ = $\frac{2x^4}{3}$  This is not correct.   Instead when we have $\int x^2 dx$ we use the power rule and come to the result:  $\frac{x^3}{3}$   In my mind this seems to ignore 'dx', so why do we even include it in the first place? Why is the integral not written as:  $\int x^2 $
$dx$ appears in differential equations, such us derivatives and integrals.  For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is.
In my Master thesis, I would like to have the title page without any page number and an acknowledgments section and the table of contents with roman numbers. The rest of the document should be numbered arabic.  I have tried this, to have roman numbers for the Table of contents and normal numbers for the rest.  \documentclass[12pt,ngerman,openany]{report} \usepackage{titlesec} \usepackage{titletoc} \usepackage{cite} \usepackage[a4paper]{geometry} %\usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage[ngerman]{babel} %\usepackage{ngerman}{babel} \usepackage[T1]{fontenc} \usepackage{graphicx} \usepackage{float} \selectlanguage{ngerman} \usepackage{amssymb} \usepackage{amsmath} \usepackage{url} \usepackage{epstopdf} \usepackage{scrhack} \usepackage{listings} \usepackage{caption} \titleformat{\chapter}{\rmfamily\huge}{\textbf\thechapter.}{25pt}{\huge\textbf} \defcaptionname{ngerman}{\lstlistingname}{Code} \defcaptionname{ngerman}{\lstlistlistingname }{Codeverzeichnis}  \lstloadlanguages{R}  \lstset{% general command to set parameter(s)    basicstyle=\sffamily\footnotesize, % print whole listing small    keywordstyle=\sffamily\footnotesize\bfseries, % ubold black keywords    identifierstyle=, % nothing happens    commentstyle=\sffamily\footnotesize\slshape, % green comments    stringstyle=\sffamily\footnotesize, % typewriter type for strings    showstringspaces=false, % no special string spaces    numbers=left,    numberstyle=\sffamily\footnotesize,    stepnumber=1,    numbersep=10pt,    showspaces=false,    showtabs=false,    frame=lines,    morecomment=[l]{\%},    float=htbp,    numberbychapter=true  }   \begin{document} \let\cleardoublepage\clearpage   \pagenumbering{roman}  \tableofcontents  \clearpage  \vspace*{75mm} \chapter{Einleitung} \section{Motivation} bla bla bla \section{Ziel}  bla bla \chapter{Stand der Technik zur Mustererkennung} \section{Die Zeitreihen} bla bla \subsection{Beschreibung} bla bla \subsection{Erklärung}  bla bla  \subsection{Vorhersage} bla bla \subsection{Kontrolle} bla bla \section{Feature Extraktion} bla bla \subsection{Autokorrelation} \subsection{Was ist Autokorrelation?} bla bla   \subsection{Autokorrelation und Zeitreihen} bla bla  \subsubsection{Zeitreihe-Diagramm}  bla bla \subsubsection{Lagged Streudiagramm} bla bla  \subsubsection{Autocorrelation-Function (Korrelogramm)} bla bla \subsection{Spektralanalyse} \subsection{Was ist Spektral-Analyse} bla bla \end{document} Evaluating $\int \frac{1}{{x^4+1}} dx$ How can I get the amount of available memory portably across distributions? The standard files/tools that report memory seem to have different formats on different Linux distributions. For example, on Arch and Ubuntu.   Arch  $ free               total        used        free      shared  buff/cache   available Mem:        8169312     3870392     2648348       97884     1650572     4110336 Swap:      16777212      389588    16387624   $ head /proc/meminfo  MemTotal:        8169312 kB MemFree:         2625668 kB MemAvailable:    4088520 kB Buffers:          239688 kB Cached:          1224520 kB SwapCached:        17452 kB Active:          4074548 kB Inactive:        1035716 kB Active(anon):    3247948 kB Inactive(anon):   497684 kB  Ubuntu  $ free              total       used       free     shared    buffers     cached Mem:      80642828   69076080   11566748    3063796     150688   58358264 -/+ buffers/cache:   10567128   70075700 Swap:     20971516    5828472   15143044   $ head /proc/meminfo  MemTotal:       80642828 kB MemFree:        11565936 kB Buffers:          150688 kB Cached:         58358264 kB SwapCached:      2173912 kB Active:         27305364 kB Inactive:       40004480 kB Active(anon):    7584320 kB Inactive(anon):  4280400 kB Active(file):   19721044 kB    So, how can I portably (across Linux distros only) and reliably get the amount of memory—excluding swap—that is available for my software to use at a particular time? Presumably that's what's shown as "available" and "MemAvailable" in the output of free and cat /proc/meminfo in Arch but how do I get the same in Ubuntu or another distribution? "Access denied" when trying to end a process on Windows 8 In a non-inertial reference frame, the centripetal force is balanced by a centrifugal force. Therefore, shouldn't the net acceleration be zero? Yet, you still feel an acceleration outwards? Why is that? Suppose $k$ is a field that is finitely generated as a ${\mathbb Z}$-algebra.  (That is, $k$ is a quotient of ${\mathbb Z}[X_1,\dots,X_n]$ for some $n$). Does it follow that $k$ is finite? GUI to CLI command translator Is there any application (or sequence of commands) for this function: seeing the result of the actions, while using a GUI interface, being automatically translated into a Terminal (or similar CLI environment) on the same machine.  I saw this in action long years ago in VBA for Applications (for example the Macros in Excel). I wonder if there is same function in Linux. Retrieve Force.com registered domain Is a current transformer appropriate for a voltage step up flyback converter? I am designing a flyback converter, input 170V output 1200V at 2mA.  I am having trouble finding an appropriate transformer, most do not have a high enough N. Current transformers seem to have ranges in 1:500, so I could work with that.  Is it proper to use a current transformer as a conventional step up voltage transformer? Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$ I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$.  I suppose I could use the counting rules in probability, perhaps combination= ${{n} \choose {r}}=\frac{n!}{r!(n-r!)}$.  I want to see an actual proof behind this equation.  Does anyone have any ideas? To prove Cayley-Hamilton theorem, why can't we substitute $A$ for $\lambda$ in $p(\lambda) = \det(\lambda I - A)$? What's the best way to develop old, exposed film? A friend of mine just found 3 rolls of exposed 135 (i.e. 35mm) film. They are at least 10 years old. He'd like to know if it is even worth taking them to the local hour photo lab but also if he should give them any special instructions. As  shows, we nearly always use the possessive form of personal pronouns for friend of mine/his/ours/etc.  But when it comes to actual names, we prefer  without the possessive apostrophe. That preference is even more marked with, say, . Not that I think the usage itself is particularly American - it's much the same with .  Personally, I find friend of him grates. In general I've no strong feelings either way as to whether it's friend of Peter or friend of Peter's (though I deplore the possessive in ), but in line with many others, I really don't like the possessive in relation to things like countries.  Why is this?   Edit: Noting an apparent "progression" (pronoun -> person -> nation) marked by reduction in use of the possessive, I checked at a finer "granularity". NGram shows that although it does occur,  virtually "flatlines" against friend of mine. But the bias reduces through of you, of us, and by the time I get to  it's much less extreme. There seems to be something "egocentric" about the double possessive.  Presumably when babies learn to speak, they soon notice that possessive pronouns, possessive apostrophes, and the word "of", all do the same job. Parents would correct a child who says "of mine's", but probably wouldn't even notice the same "redundancy" in "of Peter's". Younger speakers are unlikely to even be talking about something "of America's". Perhaps as we mature we tend to discard the "double possessive" for the more "distant" things that only adults are likely talk about, but we keep it for "closer" people because that's how we spoke when we were younger.  EDIT2 I note that I'm a great fan of him is vanishingly rare compared to ...fan of his, but with ...fan (of John) the double possessive occurs far less often than ...friend (of John's). Usage seems to be affected by the noun before "of" as well as the one after it. This is getting complicated... I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time before someone found a hole in the argument. Does this happen anymore now that we have computers? I imagine not. But it seems totally possible that this could have happened back in the Enlightenment.   Feel free to interpret this how you wish! I'm playing around with height maps (bitmaps), trying to create some of my own in my game, and for that I need to implement some basic drawing methods. I've quickly realized that drawing straight lines isn't as basic as I thought.  It simple's if your points share an X or Y coordinate, or if they are aligned so you can draw a perfectly diagonal line. But in all other cases its trickier.  What algorithm do you use to determine what pixels need to be coloured for it to become a "straight" line ? QGIS locked into fullscreen mode on Ubuntu 12.04 Decode escaped Url without using HttpUtility.UrlDecode How to pass a list of columns to data.table where some are predetermined If a drive doesn't know about partitions and files systems why moving a file from one partition to another takes relatively long time, unlike the instantaneous move on the same partition? Is that file system inefficiency? In this case, NTFS. It seems like the file is being copied and then original is erased, instead of removing the entry about that file in one partition and making it in another? Add the ability to ignore users