Why is the integral of $0.5/x = 1/2 \ln x$ and not $1/2 \ln 2x$? When you calculate $\int\frac{1}{2x}dx$ you get $\frac{1}{2}\ln(x)$ and when you calculate $\int\frac{1}{2x}dx$ you don't get $\frac{1}{2}\ln(2x)$.  $\frac{1}{2x}$ is the same as $\frac{1}{2x}$  why do you get different answers?
Two different solutions to integral Given the very simple integral  \begin{equation} \int -\frac{1}{2x} dx \end{equation}  The obvious solution is  \begin{equation} \int -\frac{1}{2x} dx = -\frac{1}{2} \int \frac{1}{x} dx = -\frac{1}{2} \ln{|x|} + C \end{equation}  However, by the following integration rule \begin{equation} \int \frac{1}{ax + b} dx = \frac{1}{a} \ln{|ax + b|} + C \end{equation}  the following solution is obtained \begin{equation} \int -\frac{1}{2x} dx = -\frac{1}{2}\ln{|-2x|} + C \end{equation}  Why are these solutions different? Which is correct?   The second solution can be simplified \begin{equation} -\frac{1}{2}\ln{|-2x|} + C = -\frac{1}{2}\ln{|-2|} -\frac{1}{2}\ln{|x|} + C= -\ln{\frac{1}{\sqrt{2}}} - \frac{1}{2}\ln{|x|} + C \end{equation} but they still differ.
Referring to correctly-styled algpseudocode procedure name in document Given an algorithm written in algpseudocode:  \documentclass{article} \usepackage{algorithm} \usepackage{algpseudocode} \begin{document} \begin{algorithm}   \caption{Part 1}   \begin{algorithmic}[1]     \Procedure {FooProc}{$G$}       \State ...     \EndProcedure   \end{algorithmic} \end{algorithm} \end{document}   How can I now refer to FooProc in my document (outside of the algorithm environment) such that it has the same typeset as in inside the algorithm environment? Shadow Incorrect at Bottom of Edge So I have been tasked to make an array without duplicated values from another existing array. So I did it, but I want to know is there some other better way to do that. Example input/output: Input: 10, 15, 10, 5, 1, 3 Output: 10, 15, 5, 1, 3 So here is my code. #include &lt;stdio.h&gt;  int main(void) {   const int MAX_ARRAY_SIZE = 5;    int m[MAX_ARRAY_SIZE], p[MAX_ARRAY_SIZE];     for(int i = 0; i &lt; MAX_ARRAY_SIZE; i++) {     printf(&quot;Enter number: &quot;);     scanf(&quot;%d&quot;,&amp;m[i]);   }   int k = 0;   int dup = 0;   for(int i =0; i &lt; MAX_ARRAY_SIZE; i++) {     for(int j = i +1; j &lt;MAX_ARRAY_SIZE; j++) {         if(m[i] == m[j]) {             dup = 1;         }     }     if(dup != 1) {       p[k++] = m[i];     }     dup = 0;   }    printf(&quot;The new array without repeated values\n&quot;);   for(int i = 0; i &lt; k; i++) {     printf(&quot;%d\n&quot;,p[i]);   }    return 0; }  I'm not sure if this is the right and simple way I do that. I want some suggestions. Thanks in advance. :) When traveling to a country with a different currency, how should you take your money? Recently my wife and I traveled to Italy and before we left we got about 85% of our money in travelers checks and the other 15% we carried on in cash. When we arrived there, no one would take our travelers checks and many places did not accept credit cards. Eventually, we found some currency exchange places that would take the travelers checks and give us Euro's, but at a 7-15% fee, which was not really a very fun surprise.   While I was under the impression that it is safer to use travelers checks, it seems that they were not really accepted (the usual explanation being fraud). So the question is, is it best to just exchange your currency at home and carry all of it with you when you travel or are there other alternatives to carrying cash that are safer and cost effective? Given that $a,b,c$ are distinct positive real numbers, prove that $(a + b +c)\big( \frac1{a}+ \frac1{b} + \frac1{c}\big)&gt;9$   This is how I tried doing it:  Let $p= a + b + c,$ and $q=\frac1{a}+ \frac1{b} + \frac1{c}$.  Using AM>GM for $p, q$, I get:  $$\frac{p+q}{2} &gt; {(pq)}^{1/2}$$ $$\sqrt{(a + b +c)\bigg(\frac1{a}+ \frac1{b} + \frac1{c}\bigg)} &lt; \frac{\big(a+\frac1{a} + b+\frac1{b} + c+\frac1{c}\big)}2$$ And for any $x\in \mathbb{R}, \space \space x+\frac1{x}≥2.$ Thus,   $$(a + b +c)\bigg(\frac1{a}+ \frac1{b} + \frac1{c}\bigg)&lt;9, $$ which is the opposite of what had to be proven. What did I do wrong? Making leaflet control open by default I'm using leaflet to display a map.  I have a control section that I'd like to be open by default, but I can't find a way to do it through the . Is there a way to do this? How to install aircrack on Ubuntu 12.04 Can humans eat grass? Can a human eat grass and digest it? 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The card is supported on the Mac once it boots, but I would like to get boot screen back since my old 8800GT has been baked in the oven a few too many times by now :-)  I've seen numerous guides on the internet about flashing the BIOS on the video cards and also ones that speak about adding the EFI BIOS, but in the haphazard forum approach, it isn't clear how this all works. To some degree, I've recently come away with the notion that you don't get rid of the existing BIOS, but rather add an EFI partition to it that the Mac would recognize. If that assumption is correct, then it would lead me to believe that adding an EFI BIOS that works on a similar card (R9 280X or 7950 which are both close but not quite) should work since I'm assuming that the existing BIOS would be similar enough to hook in. Failing that, I'm not clear how you would get a card-specific EFI BIOS for boot screen support.  Anyone an expert on this? I'm just really having a hard time trying to sift through all this and hoping to get a more direct answer! What determines speciation at a molecular level?  At what point does a scientist determine two lineages are different enough to be considered separate species?  Does it have a margin of error? What inspired Tauriel's story arc? I know Tauriel is a Jackson created character, but I wondered if anywhere in the legendarium there was fodder for the story arc between her and Kili.  I do not have access to all my books for the time being and recall stories of love between men and elves in the Silmarillion and Appendices but don't remember any such stories between elves and dwarves and wondered if there was any basis of inspiration for this aspect of their relationship within Tolkien's works.   This movie, of course, departs even further from the original than those made before it, but as a story unto itself, I can still appreciate the ride, so if there is no Tolkien basis, what inspired Jackson, Boyens, and Walsh to create this particular arc?  Is there interview info or anything? How to add existing user to an existing group? I want to add the Apache user (www-data) to the audio group. I've read the man page for useradd, but I'm not having any luck. I'm running xubuntu 11.10. Here's what I'm doing:  $ sudo useradd -G audio www-data useradd: user 'www-data' already exists   If I leave out the -G option, bash, prints the help info for useradd:  $ sudo useradd  audio www-data Usage: useradd [options] LOGIN Options: -b, --base-dir BASE_DIR       base directory for the home directory...   It's not clear to me from the man page what options I should use to make this work. Does Gollum know that others call him Gollum? How do you secure wood framing to a concrete basement floor? 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