$$\int^{ \pi /2}_{0} \ln (\sin x)\  dx$$  The answer is $- \frac{\pi}{2} \ln 2$.  I have changed it into $$\frac{1}{2} \int^{1}_{0} t d \ln t^{2}$$  But I didn't get the answer with it.
How to compute the following integral?  $$\int\log(\sin x)\,dx$$    Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of intervals where $\log (\sin x)$ is defined?
Speak Slower or Speak Slowlier? AFAIK the correct grammar for "speak slow" is "speak slowly" (slowly being an adverb). Please correct me if I am mistaken.  But in daily life I have not heard anyone saying "Speak slowlier".  I think I heard folks saying "Speak slower" but I also think it is incorrect.  Which is the best or correct form to convey the message?  Could that be the longish "Speak more slowly?" I have enabled storing Desktop &amp; Document Folders in iCloud. I keep my Xcode projects on Desktop. Every hour or so, iCloud will store files, inside of Pods folder present alongside the Xcode Project in iCloud. This is very tedious as I then have to click on each one of them one-by-one to re-download.  How can I prevent iCloud from downloading my files in this manner, for this specific Xcode project?  I have tried disabling iCloud for Desktop, but this results in me losing all my Desktop folders/files, so I could rather not go this route. How to check if a string contains a substring in Bash I have a string in Bash: string=&quot;My string&quot;  How can I test if it contains another string? if [ $string ?? 'foo' ]; then   echo &quot;It's there!&quot; fi  Where ?? is my unknown operator. Do I use echo and grep? if echo &quot;$string&quot; | grep 'foo'; then   echo &quot;It's there!&quot; fi  That looks a bit clumsy. I want to appropriate line segments that share the same geometry with a particular line width and labels based on the aggregated values of one field "GIS@BUS_5_Min".  My attribute table and polyline objects look as following, see image below   "BUSLINIE" means a bus line, "Overlap" means if the line has overlaps, i.e. it's geometry repeats.  Values of the field "GIS@BUS_5_Min" for these three objects has to be aggregated as one value and represented as a label sum("GIS@BUS_5_Min").  And the width of this line (selected in yellow) will be sum("GIS@BUS_5_Min")/100 = 19.10/100 = 0.1910  How can I achieve such output?  I assume it is possible with Layer Properties &gt; Width &gt; Data defined override &gt; Edit..., but I do not know how to check the same geometry and how to sum the values of certain features with CASE WHEN condition THEN result END conditional function?  Dissolve function does not work in my case because I do not have any key-fields.    References: Is there any way to find out if a file is a directory?  I have the file name in a variable. In Perl I can do this:  if(-d $var) { print "it's a directory\n" } The square root of a prime number is irrational If $p$ is a prime number, then $\sqrt{p}$ is irrational.  I know that this question has been asked but I just want to make sure that my method is clear. My method is as follows:  Let us assume that the square root of the prime number $p$ is rational. Hence we can write $\sqrt{p} = \frac{a}{b}$. (In their lowest form.) Then $p = \frac{a^2}{b^2}$, and so $p b^2 = a^2$. Hence $p$ divides $a^2$, so $p$ divides $a$. Substitute $a$ by $pk$. Find out that $p$ divides $b$. Hence this is a contradiction as they should be relatively prime, i.e., gcd$(a,b)=1$. dd aplied to dev USB before connect corrupt /dev/sdX and dont work Can Button[] generate output without using Print[]? How can I use QGIS to create a metro style map Is 'that is' optional? When I write something like this:     I am a man that is doing these things.   Is it correct to shorten it like this?     I am a man doing these things.   And does it require commas? This problem is really proving to be a tricky one (and quite annoying).  In SQL Server Management Studio 2008, up until a few days ago my IntelliSense was working great.  Then all of a sudden it stopped.  The icon as it as enabled on the toolbar menu, and under Tools -> Options -> Text Editor -> T-SQL -> IntelliSense it says it is enabled there.  I have tried refeshing the IntelliSense cache with Ctrl-Shft-R but that doesn't work either.  Any ideas what happened to my IntelliSense and what I need to do to get it back? How do I make Ubuntu not require a password to supend? Deploying Qt 5 App on Windows What is "it" in the following sentence: It is clear that Bob likes doughnuts I want to know how EXACTLY the WPA password gets encrypted  I have been searching through and I was able to catch:   The actual password is converted to some hash Salting is applied to the hash The operation is not reversible The hash comes with the 4-way handshake   So what kind of hash is it converted? like, SHA1? MD5? and how the salting process is done? like, is the SSID appended to the password before the hashing operation?  Or maybe I have some misunderstanding here? How do I get the local machine name in C#? Einstein-Hilbert action as an effective field theory I have read the following statement:      "The Einstein-Hilbert action can be seen as an effective field theory: setting $g_{\mu\nu}= \eta_{\mu\nu}+h_{\mu\nu}$, one gets a free Lagrangian for the field $h_{\mu\nu}$ plus interactions."   Now, how is it possible to define all the interaction terms?  EDIT: I ask apologize if my statement was appeared unclear: I have created a  which is more precise; please don't continue to consider seriously this one. Did Daenerys mess up regarding caravans? Understanding how references and labels work I'm trying to understand why I cannot effectively reference figures or equations that I have defined with labels.  For example:  \begin{figure}   \label{fig:area_Treasuries}   \caption{Treasuries as a portion of total financial flows.}   \centering   \includegraphics[width=0.75\textwidth]{graphs/Area_Treasuries.png} \end{figure}  \paragraph{} Figure~\ref{fig:area_Treasuries} presents Treasuries (both flowing into and out of the United States) as a share of total financial flows.  From 1985 to 2010, Treasuries account for, on average, 53.2\% of total financial flows.   With the above code, I get this warning:      LaTeX Warning: Reference `fig:area_Treasuries' on page 37 undefined on   input line 477.   The result is "Figure ?? presents Treasuries..."  Why is this happening?  Related, I don't understand why I can't reference, say, an equation before (I mean physically above in the text) the actual equation.  Isn't this what compiling multiple times and cross references is supposed to take care of? In what order should Tolkien's writings on Middle-earth be read?