From the definition of indefinite integral I might say: Since the derivative of a constant is zero, thus the indefinite integral of zero is a constant. Therefore: $$ \frac{dc}{dx} = 0 \quad\iff\quad \int 0dx = c, \quad\forall c\in\mathbb{R} $$  However... we know that $0\in\mathbb{R}$, and since zero is a constant, I can pull it out the integral: $$ \int 0dx = 0\cdot\int 1dx = 0\cdot(x+c) = 0 $$  And then we end up that integral of zero is zero, not an arbitrary constant. Where is wrong here?
I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$.
What's the difference between struct and class in .NET? Q1. Find the general solution to the difference equation  $$ a_{n} - 4a_{n-1} + 3a_{n-2} = 6 $$    Q2. Solve the difference equation   $$ a_{n} - a_{n-1} - 2a_{n-2} = 0, a_0 = 2, a_1 = 4 $$    I am completely lost in solving recurrence relation questions. Can anyone guide me in steps to solve the following 2 questions? Finder search doesn't work properly after Yosemite update Ever since I upgraded to Yosemite, search no longer works properly in Finder. Spotlight works fine but Finder search just gives me a blank window with a spinning ball.  I've tried reindexing and I've also tried deleting Spotlight's .plist file. I restart my MacBook Pro after doing either. Finder search will work for one or two queries and stop again. I don't know what else I can do. CodeIgniter/PHP Active Record won't increment an integer I've read and have heard of both 'as per' and 'per' being used conversationally, both with the same connotation of either 'according to' or 'on authority of'  Examples:  "Tell Ron to start molding new rollers for the mecanum wheels, as per me"  "Per Mr. Crane, you need to start the molds for the new rollers."  They both establish the same context by setting that something should be done on the authority of someone else (or the speaker, per the first example (see what I did there?)). Or it acts as a way to cite a source of a statement, directive or fact in conversational speech.  My question is, what is, if any, the functional and syntactical difference between as per and per. If there are no differences, which is apropos to use? What is the default /etc/inittab file on a fresh install of Debian Jessie? I'm looking for the full contents of the default /etc/inittab file on a fresh install of Debian 8.7 before any system-specific modifications are made.    I searched for this on the Debian website and other forums, but could not find it. Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant. How do you get a bad transcript past Ph.D. admissions? I have a master's degree in International Studies, and a double major with Computer Science from undergrad.  My transcripts suck.  There's no other way to dress it up.  I have pretty good teaching experience, and my GRE scores are awesome, and I suspect my recommendations are as bland as everyone else's.  Basically, to an admissions committee, I suspect I'm the model of a student who is probably smart enough but didn't work hard enough.  I want to do a Ph.D. in Political Science, but the response from my applications is looking pretty grim.  Am I permanently out of the running, or is there something I can do for the next few years which will help to counterbalance my unfavorable GPA? How to check if a service is running on Android? How do I check if a background service is running?  I want an Android activity that toggles the state of the service -- it lets me turn it on if it is off and off if it is on. How to wrap lines in a jtable cell? Why do the less than symbol () appear wrong as upside down exclamation (¡) or question mark (¿)? When typing &lt; or &gt; in LaTeX and compiling with pdflatex, the less than and greater than symbols appear at upside down exclamation points. I'm not in math mode. Solve $x^2+5x+6 \equiv 0 \pmod{\!11\cdot 17}$ Solve $x^2+5x+6 \equiv 187 \mod 187$ Solution $$x^2+5x+6 \equiv 187 \mod 187$$ $$ (x+\frac{5}{2})^2 \equiv \frac{1}{4}$$ $$ 4(x+\frac{5}{2})^2 \equiv 1$$ $$ y:= x+\frac{5}{2} $$ $$ 4y^2 \equiv 1 \mod 11 \wedge 4y^2 \equiv 1 \mod 17  $$ $$ ( 2y \equiv 1 \mod 11 \vee 2y \equiv 10 \mod 11 ) \wedge ( 2y \equiv 1 \mod 17  \vee 2y \equiv 13 \mod 17)  $$ $$ ( y \equiv 6 \mod 11 \vee y \equiv 5 \mod 11 ) \wedge ( y \equiv 9 \mod 17  \vee y \equiv 15 \mod 17)  $$ Combining that from CRT I got: $$ y \in \left\{49, 60,83,94 \right\} $$ and for example: $$ x+\frac{5}{2}  \equiv 94 \mod 187$$ $$ 2x  \equiv 183 \mod 187$$ some calculus and get... $$x \equiv 185 $$ And the same thing for each other case. Question Is there any faster (or smarter) way to solve equations like that? "He didn't know where New Jersey was" I know the past tense carries the past tense in every dependent clause, but referring specifically to places or to things that are eternal, like the Earth, seems a bit weird and therefore we sometimes (I believe incorrectly) say     He didn't know that New Jersey was actually on the East Coast.   Because it still is. Or     He thought the Earth was round.   So is it square now?  Logically speaking, would you consider the use of past tense here a bit confusing in a day-to-day speech in these examples? Would you instinctively opt for using the present tense? Having problems with listings and UTF-8. Can it be fixed? What are these auxiliary file descriptors? While poking around /proc/self folder for my mksh shell,  I found a peculiar thing: in /proc/self/fd/* there are all the standard file descriptors (0 for stdin, 1 for stdout, and 2 stderr) some file descriptors, but also some extra ones - 24, 25, 3. And I technically can list them with a glob in the shell:  $ for fd in /proc/self/fd/* ; do echo $fd ; done                                                       /proc/self/fd/0 /proc/self/fd/1 /proc/self/fd/2 /proc/self/fd/24 /proc/self/fd/25 /proc/self/fd/3   But when I try to stat them or use find on them, they are reported as non-existent.  $ find  /proc/self/fd/*                                                                                /proc/self/fd/0 /proc/self/fd/1 /proc/self/fd/2 find: ‘/proc/self/fd/24’: No such file or directory find: ‘/proc/self/fd/25’: No such file or directory /proc/self/fd/3   Same occurs in bash , but just with just one auxiliary file descriptor.  $ for fd in /proc/self/fd/* ; do echo $fd; done /proc/self/fd/0 /proc/self/fd/1 /proc/self/fd/2 /proc/self/fd/255 /proc/self/fd/3  $ find /proc/self/fd/* /proc/self/fd/0 /proc/self/fd/1 /proc/self/fd/2 find: ‘/proc/self/fd/255’: No such file or directory /proc/self/fd/3   Question is: what are those additional file descriptors? What is their purpose? What do I need in order to create an application for ubantu? I tried to learn Dvorak keyboard layout a few years ago, and got reasonably good at it, but I couldn't get past the fact that command-key shortcuts were no longer in the right place.  In particular, Ctrl+Z (undo), Ctrl+X (cut), Ctrl+C (copy), and Ctrl+V (paste) are all chosen specifically because of the location of those keys on a Qwerty keyboard (well, maybe Ctrl+C makes sense).  Ctrl+A (select all) and Ctrl+S (save) are also very convenient because of their Qwerty keyboard location.  Is there any way (on Windows XP) to use Dvorak except when a control key (Ctrl, Alt, and/or WinLogo) is pressed?  (Preferably without having to install any third party app.) Rendering an animation the way 3D view looks Is there a device that could measure the speed of time? Is there (or can there be) a device that could measure the speed and acceleration of time? Why is there not a simpler way to calculate the standard deviation? Steps of getting standard deviation. :        Work out the average (mean value) of your set of numbers   Work out the difference between each number and the mean   Square the differences   Add up the square of all the differences   Divide this by one less than the number of numbers in your set -   this is called the variance   Take the square root of the variance and you've got the standard   deviation      Am I missing out something, or why do we need to square the differences in step 3?  Why not simply do a Abs (multiply all negative numbers by -1) in step 3?  Also, my second question is why do we need to divide by one less than the number of numbers in the set in step 5? why not simply divide by the number of numbers?