What is $$\int \frac{1}{\sqrt{25-x^2}}$$ WolframAlpha says $\sin^{-1}(\frac{x}{5})$ while I got $\frac{1}{5}\sin^{-1}(\frac{x}{5})$. What is correct? Thanks in advance.
Somewhere in the provided answer:   $$\int \frac{1}{\sqrt{2-x^2}} dx = \sin^{-1}{\frac{x}{\sqrt{2}}}$$  How did they get that? What I have:   $$\frac{1}{\sqrt{2-x^2}} = \frac{1}{\sqrt{2(1-\frac{x^2}{2})}} = \frac{1}{\sqrt{2} \sqrt{1-\frac{x^2}{2}}}$$  $$\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{1-(\frac{x}{\sqrt{2}})^2}} = \frac{1}{\sqrt{2}} \sin^{-1}{\frac{x}{\sqrt{2}}}$$  So I have an extra $\frac{1}{\sqrt{2}}$ ... I probably had some stupid mistakes?
addplot3[surf] creates lines instead of a surface I have been trying to create a surface plot but I am unable to get it to work. Any help would be greatly appreciated. I've tried using surf which only produces lines instead of a surface (Fig. 1). Note: Although I was originally able to produce output, thesurf plot no longer compiles: "Package tikz Error". I've tried compiling with both pdflatex and lualatex with -shell-escape flag.    Figure 1  I also tried to use the patch style plot. This produces gibberish (Fig. 2)    Figure 2  MWE:  \documentclass{article} \usepackage{pgfplots} \usepgfplotslibrary{external} \usepgfplotslibrary{patchplots} \tikzexternalize  \begin{document} \begin{figure}[t]   \centering    \tikzsetnextfilename{surface_plot}   \begin{tikzpicture}     \begin{axis} [       height=12cm,       width=15cm,       grid=major,       grid style={dotted},       colormap/jet,       mesh/ordering=x varies     ]       %\addplot3[patch,patch type=biquadratic,shader=faceted interp,patch refines=3] coordinates {       \addplot3[surf] coordinates {  (-6,-6,446.788)  (-5,-6,423.361)  (-4,-6,413.264)  (-3,-6,402.349)  (-2,-6,388.948)  (-1,-6,391.747)  (0,-6,386.874)  (1,-6,423.778)  (2,-6, 439.19)  (3,-6,449.074)  (4,-6,462.155)  (5,-6,484.399)  (6,-6,505.412) (-6,-5, 423.48)  (-5,-5,392.382)  (-4,-5,406.696)  (-3,-5,392.592)  (-2,-5,371.158)  (-1,-5,384.142)  (0,-5,394.666)  (1,-5,408.105)  (2,-5, 422.51)  (3,-5,420.173)  (4,-5,445.816)  (5,-5,464.731)  (6,-5,485.349) (-6,-4,402.646)  (-5,-4,371.879)  (-4,-4,376.239)  (-3,-4,396.874)  (-2,-4,382.761)  (-1,-4,386.665)  (0,-4,397.148)  (1,-4,410.875)  (2,-4,425.998)  (3,-4,385.143)  (4,-4,420.458)  (5,-4,438.432)  (6,-4,    458) (-6,-3,423.219)  (-5,-3,370.582)  (-4,-3,464.786)  (-3,-3,443.921)  (-2,-3,425.196)  (-1,-3,427.138)  (0,-3,438.002)  (1,-3, 453.97)  (2,-3,473.021)  (3,-3,418.583)  (4,-3,435.887)  (5,-3,414.993)  (6,-3,475.148) (-6,-2,401.995)  (-5,-2,385.612)  (-4,-2,438.956)  (-3,-2,418.583)  (-2,-2,400.093)  (-1,-2,400.773)  (0,-2, 410.58)  (1,-2,426.024)  (2,-2, 444.92)  (3,-2,397.131)  (4,-2,413.739)  (5,-2,431.741)  (6,-2,451.267) (-6,-1,382.978)  (-5,-1,335.274)  (-4,-1,416.598)  (-3,-1,396.718)  (-2,-1,378.348)  (-1,-1,377.455)  (0,-1,32.0089)  (1,-1,400.683)  (2,-1,419.918)  (3,-1,377.371)  (4,-1,393.286)  (5,-1,374.648)  (6,-1,429.077) (-6, 0,365.902)  (-5, 0,350.965)  (-4, 0,397.527)  (-3, 0, 378.36)  (-2, 0,360.454)  (-1, 0,358.447)  (0, 0,    363)  (1, 0,381.147)  (2, 0,401.177)  (3, 0,362.155)  (4, 0,378.174)  (5, 0,395.513)  (6, 0, 414.34) (-6, 1,372.511)  (-5, 1,326.945)  (-4, 1,407.215)  (-3, 1,388.646)  (-2, 1,371.945)  (-1, 1,371.003)  (0, 1,380.027)  (1, 1,396.414)  (2, 1,416.602)  (3, 1,374.897)  (4, 1,391.276)  (5, 1, 373.32)  (6, 1,428.284) (-6, 2,391.149)  (-5, 2,376.191)  (-4, 2,406.941)  (-3, 2,369.633)  (-2, 2,393.583)  (-1, 2,394.184)  (0, 2,404.263)  (1, 2,420.588)  (2, 2,  440.5)  (3, 2,416.631)  (4, 2,374.948)  (5, 2,391.855)  (6, 2,418.511) (-6, 3,375.967)  (-5, 3,361.698)  (-4, 3,357.421)  (-3, 3,368.273)  (-2, 3,377.106)  (-1, 3,378.664)  (0, 3,369.175)  (1, 3, 382.42)  (2, 3,397.821)  (3, 3,377.942)  (4, 3,384.944)  (5, 3,387.862)  (6, 3,424.429) (-6, 4,396.478)  (-5, 4,381.554)  (-4, 4,376.162)  (-3, 4,388.843)  (-2, 4,375.881)  (-1, 4,378.384)  (0, 4, 355.14)  (1, 4,403.654)  (2, 4,419.779)  (3, 4,398.599)  (4, 4,407.113)  (5, 4,426.715)  (6, 4,448.245) (-6, 5,409.364)  (-5, 5,380.492)  (-4, 5,387.843)  (-3, 5,360.034)  (-2, 5, 362.23)  (-1, 5,364.909)  (0, 5,358.321)  (1, 5,363.226)  (2, 5,379.325)  (3, 5,397.152)  (4, 5,410.086)  (5, 5,433.909)  (6, 5,460.548) (-6, 6, 423.99)  (-5, 6,408.967)  (-4, 6,400.475)  (-3, 6,372.922)  (-2, 6,371.116)  (-1, 6, 373.92)  (0, 6,384.778)  (1, 6,  398.8)  (2, 6, 415.62)  (3, 6,420.127)  (4, 6,436.064)  (5, 6,461.957)  (6, 6,491.123)        };     \end{axis}   \end{tikzpicture}      \end{figure} \end{document}   Thank you so much. Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion.  If $F: P(X) \to  P(X)$ is monotone (order preserving), then $F$ has a fixed point.  How are we going to prove that without using the term (complete lattice)?! PhD Student : Publish Paper with Wife? C++ handling very large integers I am using the RSA Algorithm for encryption/decryption, and in order to decrypt the files you have to deal with some pretty big values. More specifically, things like   P = C^d % n   = 62^65 % 133   Now that is really the only calculations that ill be doing. I have tried using Matt McCutchen's BigInteger Library, but I am getting a lot of compiler errors during linking, such as:  encryption.o(.text+0x187):encryption.cpp: undefined reference to `BigInteger::BigInteger(int)'  encryption.o(.text+0x302):encryption.cpp: undefined reference to `operator&lt;&lt;(std::ostream&amp;, BigInteger const&amp;)'  encryption.o(.text$_ZNK10BigIntegermlERKS_[BigInteger::operator*(BigInteger const&amp;) const]+0x63):encryption.cpp: undefined reference to `BigInteger::multiply(BigInteger const&amp;, BigInteger const&amp;)'   So I was wondering what would be the best way to go about handling the really big integers that come out of the RSA Algorithm.  I heard that a possibility would be to declare your variables as a double long, so...  long long decryptedCharacter;   but I'm not sure exactly how big of an integer that can store.    Well for example, I try to compile and run the following program using dev C++:  #include iostream  #include "bigint\BigIntegerLibrary.hh"  using namespace std;  int main() {     BigInteger a = 65536;     cout &lt;&lt; (a * a * a * a * a * a * a * a);     return 0; }   then I get those errors.  Derek, I thought that by including the BigIntegerLibrary.hh file, that the compiler would go through and compile all the necessary files that it will use.  How should I try and compile the program above in order to resolve the linking errors? I have a simular issue to the question asked in :  I have tried the solution provided by Jake. This makes that my picture fails to resize (scale) as i intend.  Whether i give Set scale=0.4 or Set scale=0.8 The size of the  picture remains the same  This is the code i use:  \begin{figure}[ht!] \centering \begin{tikzpicture}[Set scale=0.4, every node/.append style={transform shape},] % STYLES \tikzstyle{every node}=[rectangle,rounded corners, minimum width=100pt,     align=center,node distance=3cm,fill=black!10,inner sep=5pt,text width=3cm,minimum height=1.75cm,&gt;=stealth,text badly centered] % Draw forces \node [force, above of=rivalry] (top) {Firm strategy, structure and rivalry}; \node [force, left=1cm of rivalry] (left) {Factor conditions}; \node [force, right=1cm of rivalry] (right) {Demand Conditions}; \node [force, below of=rivalry] (bottom) {Related and Supporting industries}; %%%%%%%%%%%%%%%% % Draw the links between forces \path[&lt;-&gt;,thick]  (left) edge (right) (left) edge (top) (left) edge (bottom) (top) edge (right) (bottom) edge (right); \path[&lt;-&gt;,thick]  (bottom) edge (top); \end{tikzpicture} \caption{Porter's Diamond source:~\cite{Porter:1980}}% \label{fig:diamond}  \end{figure} Solving $\sin x\cos3x+\cos x\sin3x=\sqrt{3}/2$ in the interval $[0,2\pi]$ Solve the following trig equation: $$\sin(x)\cos(3x)+\cos(x)\sin(3x)=\frac{\sqrt{3}}{2}$$ in the interval $[0,2\pi]$.  Using trig identities, it is now at $$\sin(x+3x)=\sqrt{3}/2 \tag{1}$$ Next, it has become $$\sin(x+3x)=\sin 60^\circ \tag{2}$$ leading to $$\sin(x+3x)-\sin60^\circ=0 \tag{3}$$ From there, it is unclear what to do. I have just installed 12.04 LTS on my laptop and I'm kinda new in this. I am required to write some program in Fortran 77 and I need a Fortran 77 compiler. I have tried installing g77 as described in the Ubuntu website but was unable to do so. I have typed in the following in terminal:  sudo apt-get install g77   and get the following:  Reading package lists... Done Building dependency tree        Reading state information... Done Package g77 is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'g77' has no installation candidate   May I know what is the problem? I have tried searching g77 in the software center but apparently it is not there. Does this mean it is not available for 12.04?  I'm currently using an Asus K43S laptop with Intel Core i5-2450M processor and my Ubuntu is installed alongside Windows. Assign multiple keyboard shortcuts for one action I am creating an animated sprite list in Adobe Illustrator. I finished creating the sprites and used the Horizontal/Vertical Distribute Center commands to distribute the objects equally in some area.  But, if the objects are spread too far apart, the Distribute Center tool leaves, albeit equal, much distance between the objects. The opposite is true: if the objects are too close then distributing them will just overlap them.  My question is, is there any tool that just puts objects side by side?  I imagine that this tool would work if the objects aren't even equal in size.   I have viewed  and answer but it is not what I need. Is there a way to fix a small chip in a wall's paint? I'm not sure exactly with what I've bumped my wall and caused this this:    Is there anything that can be done to fix this without repainting the entire room?  I've been reading around a bit, and not sure exactly what to call this, hence the weird question title.  The paint is a beige-ish colour, not exactly white. The hole is about 2 x 1.5 cm in size.   I've read a bit about spackling paste, though I understand that's for small cracks and holes.  Here are some picture with a better resolution:  and   Unfortunately, I don't have much experience with this sort of thing. Help please.  Also, this is in the UK, so you can tell what stores I might have access to for materials. There are $25$ horses. You can take $5$ horses at a time and race them. Each horse always finishes the race in the same amount of time, and there are no ties. The only information you get from each race is the order that the $5$ horses finished in. What is the smallest number of races you need to find the $3$ fastest horses, in order?   I found this problem on  and cannot figure out the best answer. Please help. While any of the other sections under &quot;Edit Profile &amp; Settings&quot; do work normally. How bad is it to use display: none in CSS? I've heard many times that it's bad to use display: none for SEO reasons, as it could be an attempt to push in irrelevant popular keywords. A few questions:   Is that still received wisdom? Does it make a difference if you're only hiding a single word, or perhaps a single character? If you should avoid any use of it, what are the preferred techniques for hiding (in situations where you need it to become visible again on certain conditions)?   Some references I've found so far:       If you're straight-out using CSS to hide text, don't be surprised if that is called spam. I'm not saying that mouseovers or DHTML text or have-a-logo-but-also-have-text is spam; I answered that last one at a conference when I said "imagine how it would look to a visitor, a competitor, or someone checking out a spam report. If you show your company's name and it's Expo Markers instead of an Expo Markers logo, you should be fine. If the text you decide to show is 'Expo Markers cheap online discount buy online Expo Markers sale ...' then I would be more cautious, because that can look bad."   And  on the same article     We can flag text that appears to be hidden using CSS at Google. To date we have not algorithmically removed sites for doing that. We try hard to avoid throwing babies out with bathwater.   Eric Enge      The legitimate use of this technique is so prevalent that I would rarely expect search engines to penalize a site for using the display: none attribute. It’s just very difficult to implement an algorithm that could truly ferret out whether the particular use of display: none is meant to deceive the search engines or not. The mysql extension is deprecated and will be removed in the future: use mysqli or PDO instead The advantages and disadvantages of using Kaiser Rule to select the number of principal components Show that $\int_0^\pi f(\sin x)\,\mathrm{d}x = 2\int_0^{\pi/2}f(\sin x) \, \mathrm{d}x$ I was playing modern with the owner of a shop.  He played a  which he used to bring in 7 lands: 5 basic mountains and 2  cards.  He then claimed that each  triggered 5 times (killing me).  I am doubting this now.    Here is my reason: The two  cards entered at the same time as the other 5 basic lands.  That means when they were on the battlefield and their ability was in effect, the other 5 basic lands were already there too.  That seems too late to trigger a "Enters the Battlefield" effect.  So here is my question.  Am I right or wrong?  If I am wrong, please show me how the enter the battlefield effect can trigger for something that enters the battlefield at the same time as itself.  If I am right, please show me the rules that prove it.  I now have two shop owners that are convinced that this is a legal play.  If it is not legal, I am going to need clear rules citations to convince them.    Side note:  If I am wrong and it is a legal play, then I have a follow up scenario.  Say I have 2  cards and 5 other human cards in my graveyard.  If I play a spell that lets me return all of them to the battlefield at the same time (say ), would the  cards get 6 +1/+1 counters on them?    To me, if the trick that was played on me with  is legal then my  cards should also get the +1/+1 counters. TSA lock reset button stuck and now lock button is also stuck I just bought an Xbox 360 game with wrong region. What should I do? I just bought a  game (Forza 4) for my Xbox 360 (bought in the US). When I tried to play the game, it wouldn't load because of region issue. Is there a legal way to solve this (no crack)? There's a story my grandpa use to tell me when I was younger, about the sole survivor of of an attack on a mining vessel in space. The man lives in space for months/years, ignored by the company he worked for and never rescued. Then when he thinks he is being rescued, it's actually another set of raiders who attack him and kidnap him.  He then wakes up in captivity in some sort of prison inside of an asteroid, and has a word tattooed on his forehead. My grandpa called him Ronin (that's what the tattoo on his forehead said) but I don't know what the character's real name is. Anyways, he learns that he has the ability to teleport after having a brush with death and decides to make his way back to Earth so that he can exact revenge on the mining company who abandoned him and left him floating to die.  Does anyone know what this story or book may be? My grandpa told it to me when I was little, he said that he got it from a book he use to love reading when he was younger. I'm assuming the book is very old (possibly pre 1940's as he was born in the 20s). I can't ask him as he is no longer alive. I'm also unsure as to whether or not all the details in the story he told me, are the same as in the original.