$$\int_0^{\sqrt{5}} \sqrt{4x^4 + 16 x^2} dx$$  The square root really got me confused here. I've tried using the standard trick with $x = x + \sqrt{x^2+4}$ but failed. That might still be the best approach.  What's the best way to solve this and how to do it?
$\displaystyle \int\nolimits_0^8 \sqrt{x^4+4x^2}\,dx$.   Alright, so I thought I had this figured out. Here's what I did:   I factor out an $x^2$ to get $\sqrt{x^2(x^2+4)}$. I let $x = 2\tan(\theta)$, therefore the integrand is $\sqrt{4\tan^2(\theta)  (4\tan^2(\theta) + 4)}$. Factor out a 4 and it becomes $\sqrt{(16\tan^2(\theta)  (\tan^2(\theta) + 1))}$ Which equals $\sqrt{16\tan^2(\theta) \sec^2(\theta)}$ This is easy to take the sqrt of. The integrand becomes $4\tan(\theta)\sec(\theta)$. Now, the integral of this is $4\sec(\theta)$ And it's evaluated from $0$ to $\arctan(4)$ right? Because as $x$ goes to $0$, so does $\theta$, and as $x$ goes to $8$, $\theta$ goes to $\arctan(4)$... But the end result $(4 (\sec(\arctan(4)) - 1) )$ isn't the correct answer   I put it into WolframAlpha and I get $(8/3) (17\sqrt{17} - 1)$, which is the right answer. How did they get that? (there's no "show steps" option)  Any help is greatly appreciated!  PS, what's the syntax for doing sqrts and exponentials?
I read in a book &quot;Physics - Resnik and Halliday&quot; the explanation of Type-I Superconductors {cold ones} that:  The Electrons that make up current at super-cool temperatures move in coordinated pairs. One of the electrons in a pair may electrically distort the molecular structure of the superconducting material as it moves through, creating  nearby short-lived regions of positive charge.the other electron in the the pair may be attracted to the positive spot. According to the theory the coordination would prevent them from colliding with the molecules of the material and thus would eliminate electrical resistance  Is this the only explanation or can somebody give me a more intuitive explanation that also takes into the problem of defect scattering as in the case of resistance and also explains the Type-II superconductors {hot ones} P.S. What are &quot;coordinated pairs&quot;? Drag and drop not working inside Nautilus on Ubuntu 17.10 I'm having a strange problem on Ubuntu 17.10: when I try to drag and drop something inside Nautilus it simply doesn't work, it doesn't even select the item, it seems like it's ignoring the action... Also, I cannot select things inside nautilus by drawing rectangles with the mouse.  All of these actions work only on the Desktop. Here is a demo:    I'm using the Xorg session with nVidia proprietary drivers. I also noticed that Nautilus seems to hang when I try to copy, paste or cut a file...  EDIT 1: I solved the problem for the slow copy/paste by disabling the clipboard sync plugin of KDEConnect. I also discovered that if I open Nautilus as root, I can drag and drop or select whatever I want... We are supposed to prove that the zero vector is unique with the additive identity $x+O=x$ and then by creating another additive identity with $O'$, then $x+O'=x$.  We set them equal $x+O=x+O'$ and use the vector cancelling theorem to get $O=O'$, but how does this prove that the zero vector is unique? A cyclic group is a group that is generated by a single element.  That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$.  This element $g$ is the generator of the group.  Is that a correct explanation for what a cyclic group and a generator are? How can we find the generator of a cyclic group and how can we say how many generators should there be? As the question states... In Windows 7 with a single screen, you can drag a window to the right and a window to the left to maximize them both at half the screen, letting you quickly and neatly see two windows simultaneously.  If I have 2 monitors, is there a way to do this with 4 windows, where each monitor gets 2 windows split across each screen? What happens if you increment an iterator that is equal to the end iterator of an STL container I'm using LyX on Linux Mint 14 ...  Using BibTeX with the apalike style and some modifications in the preamble, I've managed to get the bibliography as it should be except for one thing. When I cite a work with multiple authors it gets printed with the separator "and" which isn't correct for my language (Danish). Curiously it's displayed correctly in the user interface of Lyx - for example "(Wittgenstein og Waismann 2003 - which led me to believe everything was ok but in the pdf the separator "and" is used both in the reference and in the bibliography.  Is there any way to fix this using BibTeX, or can it only be done with biber 1.5, as suggested in the last comment here:   I really hope their is because getting biber to work using the following guide has thus far proven to be impossible:   Help much appreciated ... I really hope I won't have to relapse into setting my master thesis up in LibreOffice. Chat reputation in left bar seems different than any other reputation displayed for users I need to execute a directory copy upon a user action, but the directories are quite large, so I would like to be able to perform such an action without the user being aware of the time it takes for the copy to complete.  Any suggestions would be much appreciated. Possible Duplicate:          My reputation score has gone down, but the reputation tab in my profile does not indicate why. Is this a bug? What else could have caused it? Find the all possible real solutions of $x^y=y^x$ Find the all possible real solutions of $$x^y=y^x$$ $x,y$ both are real numbers.  My attempt:I observed the following solutions  $x=2,y=4$  $x=4,y=2$  $y=x$  Is there any other possible solutions? Proving that an additive function $f$ is continuous if it is continuous at a single point Suppose that $f$ is continuous at $x_0$ and $f$ satisfies $f(x)+f(y)=f(x+y)$. Then how can we prove that $f$ is continuous at $x$ for all $x$? I seems to have problem doing anything with it. Thanks in advance. Distribution of $-\log X$ if $X$ is uniform. For $X$ and $Y$ random variables; $X$ follows the uniform distribution.  (1): if $Y=-\log X$  (2): then it can be shown that $-\log X$ is distributed as $\exp(1)$ {i.e. exponential with mean 1}.  Why is this so? Intuitively statement (2) make sense to me. But i'd like a mathematical proof.  -Probably wrong working:  (1) seems to imply $\exp(-Y) = X$ (which is like saying $X$ is exponentially distributed, which is a contradiction, since its actually uniform!); or is it incorrect for me to do this since $X$ and $Y$ are random variables?  Ultimately how do I prove (2)?  Thanks How can I give other drives and partitions short, meaningful names in nautilus? In nautilus and Unity the external drive has a nice short descriptive name "mcat", but all partitions on the 4 internal drives are displayed as a size (834 GB filesystem) or a huge 32-character string form of a GUID:  I'm guessing the external drive is nice, short, sweet and readable because that drive may have no partitions (well, just one I guess) and that name may be the drive label, whereas partitions usually don't have names. That may explain my problem, but doesn't solve it.  Is there some way to give reasonable names to these partitions in nautilus and Unity? Consider these two ways of coining a name: 1) Bolzano's theorem, Rolle's theorem 2) the Poincare conjecture, the Kantorovich theorem. What is the difference between these ways, why do people choose one over the other?  My guess is that the main difference is when you put 's at the end of a name, you strictly imply that the theorem/conjecture was written by this very person (Bolzano wrote Bolzano's theorem). But when it is "the [Name] theorem/conjecture," it doesn't mean that the person which name is used actually created this thing, maybe he just inspired it (Poincare might have only inspired the Poincare conjecture but did not propose it).  Sometimes there is only one way: ; you can not call it hairy ball's theorem. Is "data" treated as singular or plural in formal contexts? My non-native English speaking friend just asked me: "Data is..." or "Data are..."?  I said both but that's because I've been desensitized from reading/writing both (especially from writing code and adding quick comments).  My question: Is it acceptable to utilize either for a university paper? Or is one safer than the other (when confronted with stickler professors)?    Related questions: A required field that isn't really required? ...the email field in my clients data entry system is required but users of the system are allowed to enter na@na.com if the customer refuses or doesn't have an email address. Is this best practice? Have any suggestions or workarounds for their solution? Can't format my usb drive. I have already tried with mkdosfs and gparted I have a MicroSD card in a USB adapter (which plugs into a USB port on my machine, and acts like a USB flash drive). I was using Unetbootin to make this a bootable USB flash drive with Kubuntu. But I needed to cancel while it was working. So I killed the Unetbootin process from the console.  Since then, I can't format the MicroSD or delete the folder that Kubuntu made. I have also tried mkdosfs, with no success. I can't mount the drive anymore either? What can I do to make this drive usable again?  Trying to create a new partition table in GParted, as described in Eliah Kagan's answer, does not work. It fails with the error message "imposible crear tabla de particiones" ("unable to create a partition table"). Is this an abnormal rate of spoke breakage? I just broke a spoke and this is the fourth spoke I've broken in the last 12 months.  This one was very puzzling because it came at the very end of an easy 30 mile (48 km) ride.  I was pedaling very slowly into my driveway when it snapped.   All four have been on the drive side of the rear wheel.  This is a Specialized Allez, one year old, with factory wheels.  I weigh 175 lbs (79 kg) and ride about 2000 miles (3218 km) per year.  Just recreational riding on good road surfaces.  Is this an abnormal rate of breakage?    And if it is an abnormal rate of breakage, what should I do about it?  EDIT:  I took the wheel into the LBS today. They didn't seem surprised by 4 broken spokes in a year, but everyone else is, so I'm going with everyone else. The mechanic checked the tension and they were all correct with one slight exception, so that's not the issue. The wheels are factory wheels and not exactly top flight. The spokes are 1.2 mm rather than 2 mm, so pretty thin. And they're 28 spokes per wheel, so that's cutting it even thinner from what I can gather.   Rebuilding the wheels is $125 each, so that's just not worth it. I can find much better wheels for not much more, so I'm on Amazon looking at wheels while trying not to deplete my life savings. (What some people will pay for wheels boggles the mind.)  I'll have them fix the spoke and check the tensioning so I can sell them and ride until I get replacements.   Gotta buy the tools and learn to replace spokes and true wheels.... Poisson / exponential distribution Next weekend you will be participating in 12km cross country race on a mountain.The average time between  two successive wild animal sightings on the mountain is reported to be 5 minutes  (a) What is the probability that you see at least one wild animal in the 11th minute of the     race given that you will see 3 wild animals since the start of the race?  (b) What is the probability that it will take more than a quarter of a hour before you see a wild animal after ten minutes of running?  Now I have attempted (a) , but I don't know whether my thinking is correct. (b) on the other hand makes little sense to me.  My attempt:  (a)$ X~ Poisson(\frac{1}{5})$ and $Y~Exponential(\frac{1}{5})$  $Pr(X&gt;4) = Pr(X&gt;1)$  (I am thinking that this is some variation of the memoryless property)  $Pr(X&gt;1) =Pr(Y&lt;1)$  $Pr(Y&lt;1)= 1 - (e^{(\frac{-1}{5})})$  Perhaps, I was a bit to vague in my attempt of (a). So here goes my attempt of(b).From my understanding of the question , it is asking what it the probability that the time between events(in this case animal sightings) is more than 25 minutes given you have been running for ten minutes  Now from what I know the fact that you have been running ten minutes is irrelevant this is due to memory-less property of the exponential  distribution   so without further ado  I present my attempt at (b)   let $X $ be exponentially distributed random variable with $\lambda = 1/5$  then  $Pr(X &gt;15) = 1 - Pr(X &lt;= 15)$