Solve this integral$$\int_{0}^{\pi}{\log(\sin x)}\,\mathrm dx$$   I have tried this by breaking the limit from $0$ to $\pi/2$ and $\pi/2$ to $\pi$, but I am unable to solve the 2nd part.
From this paper:     Shouldn't $du$ be $dt$? And also how do you get from that line to the final result if $du$ is not $dt$?
Run Metro application on startup? I want to know where the Metro applications (Mail, People, Messages, ...) executable file path is.  Also, how can I run Mail application when my PC starts? So I was in the middle of making a ball-shaped smoke, but the volumetric effect didn't work following the basic shape. It remains cube in Eevee Blender 2.8; while in Cycle it works, but in Cycle it takes thousands of years for Render. Why WordPress choose data serialization over json_encode? In my little age with WordPress, I've seen WordPress itself and its friendly plugins are using PHP serialize() in storing data into db in many cases. But in a recent search I found a serious community support for the json_encode() over the serialize().     And I personally tested an associative array with both of 'em, that shows:  serialize() stores 342 chars json_encode() stores 285 chars  Why I'm asking this? I'm on a project while I'm going to store repeating meta fields to a post. Where:  Data would be basically in English, but sometimes can be Bengali Data would be associative array, 3 level deep (I hope I understood levels correctly):   array(     1 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ),     2 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ) )  I've checked the postmeta table's meta_value field it's a longtext,  a length of 4,294,967,295 chars (4GB). So I need a robust solution into storing things. I'm new to Linux but not to app development. I come from a RealBasic on a Mac background.   I've accepted that I need to learn a new language to develop for Linux (since RB won't run on 64 bit distros and is a cloed proprietary language) but I want to make sure I pick the right one.   I've been looking at Python and was all set to go until I read that PyGTK is becoming obsolete with the introduction of GTK 3. Is this true? The Ubuntu developer website stills seems to recommend it with Glade for the UI but I want to make sure that I don't learn something that's on the way out! Singular or plural verb with two subjects Possible Duplicate:          I'm writing an interrogative sentence questioning someone else's writing:  "Is grammar and spelling correct?"  Can I use the singular verb "is" or must it be the plural verb "are"?  (The sentence is one of a list of interrogative questions about the quality of writing, e.g. "Are words in the right order?" "Does the copy flow?") As homework, I have to prove that     $\forall n \in \mathbb{N}: n^3-n$ is divisible by 6   I used induction  1) basis: $A(0): 0^3-0 = 6x$ , $x \in \mathbb{N}_0$ // the 6x states that the result is a multiple of 6, right?  2) requirement: $A(n):n^3-n=6x$  3) statement: $A(n+1): (n+1)^3-(n+1)=6x$  4) step: $n^3-n+(n+1)^3-n=6x+(n+1)^3-n$  So when I resolve that I do get the equation: $n^3-n=6x$ so the statement is true for $\forall n \in \mathbb{N}$  Did I do something wrong or is it that simple? Input file1 is:  dog 123 4335 cat 13123 23424  deer 2131 213132 bear 2313 21313   I give the match the  pattern from in other file ( like dog 123 4335 from   file2).  I match the pattern of the line is dog 123 4335 and after printing all lines without match line my output is:   cat 13123 23424 deer 2131 213132 bear 2313 21313   If only use without address of line only use the pattern, for example 1s how to  match and  print the lines? I'm more of a math guy than a physics guy so bear with me....  In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal dimension of around $1.2$. You can also define a concept called fractal extent (which is the experimental version of hausdorff measure). This extent is a measure of how large a fractal is. It's even been used to more accurately model the distribution of coastline bird nesting.   Even though fractals work as an interpretation of geometrical units, I don't think they would describe things such as $kg^{1/2}$ very well. In addition, I've heard about white Gaussian noise that has square root time units.   What is the physical meaning of fractional units such as the above? (Since it obviously has little geometric meaning) When you are performing time consuming operations in bash like installing new software, is it possible to write text that will appear at the prompt when the operation has finished.  E.g   Imagine running:  apt-get install eclipse-platform   Then it will use quite a while to finish while you see the installation log, meanwhile I want to create new folders (workspaces) which Eclipse later will use.  Is this possible without opening a new terminal (or tab)? How do you make Ubuntu accept files sent over bluetooth I have a problem when I send my files from phone to ubuntu 12.04 via bluetooth as the phone prompts with the message "the file not sent"! I can send any files from ubuntu 12.04 to my android phone. Did other meta-humans enter a coma? In the first episode of The Flash (2014), Barry's struck by lightning, some chemicals fall on him, and he enters a coma for 9 months.  This happens when the S.T.A.R. Labs particle accelerator is turned on for the first time.  9 months later, he wakes up and has powers.  He then starts fighting crime, especially other meta-humans who were created by the same particle accelerator accident.  However, this is all 9 months after the accident, where were the other meta-humans during those 9 months?  The S.T.A.R. Labs people make it sound like they hadn't seen any meta-humans active during the 9 months Barry was out.  Did all people granted powers by the particle accelerator accident also fall into comas?  Why weren't they active heroes/villains during the 9 month gap? The school where I work has an S when it is an acronym (NEGS) it looks odd with an apostrophe to show ownership e.g. NEGS' tennis team...any thoughts? Solve for $x$ in $6x^2-25x+12+\frac{25}{x}+\frac{6}{x^2}=0$ I simplified to get $6x^4-25x^3+12x^2+25x+6=0$  Or $6 (x^2+1)^2+25x(1-x^2)=0$  Then I substituted $z=x^2+1$, to get $6z^2+25\sqrt{(z-1)}(2-z)=0$  I can't find a way to proceed further. How to remove Apple System Apps How can I delete Apple OS X application such as Photos, iPhoto, and others that I never use? I've just made a set of armor by using a male body mesh model I downloaded and selecting certain faces from the model and duplicating them to create viable armor pieces.   Since the pieces are just duplicated from the same mesh male model, then it's still part of the same object.   How do I separate the armor piece from the model to make the pieces separate? Can open, unsafe nuclear fusion reaction burn the atmosphere? I happened to hear people saying that the nuclear fusion bomb tests could set the atmosphere on fire. I have some serious doubts about that - but I have no facts. Nuclear fusion reaction requires $15*10^{6}$ kelvins to start. If we produce such temperature in "open air" would the atmosphere become a fuel for further fusion? Shouldn't the whole thing just be torn apart by its terrible pressure? It seems that OTP encryption is unbreakable. I have read some articles (Wikipedia, Blogpost, etc.), but I could not find any information about how they are used.  If I wanted to create a messenger service, how would I distribute the OTP keys? Both the sender and the receiver need this key before the message is sent, right?   And how do I make sure no one else gets these keys? Setting up PyCharm to reference ArcPy? How would I setup  to recognize import arcpy?   I'd like to use PyCharm to develop and debug geoprocessing tool. Does installing and using Wine open up your Linux platform to Windows viruses? Just wondering if installing Wine might open up a fairly solid Linux desktop to the world of Windows viruses.  Any confirmed reports about that?  Would you then install a Windows antivirus product under Wine? I'm looking for an algorithm which when given a latitude and longitude pair and a vector translation in meters in Cartesian coordinates (x,y) would give me a new coordinate. Sort of like a reverse Haversine. I could also work with a distance and a heading transformation, but this would probably be slower and not as accurate. Ideally, the algorithm should be fast as I'm working on an embedded system. Accuracy is not critical, within 10 meters would be good.