$$\int \dfrac{\sqrt{\sin^4x+\cos^4x}}{\sin^3x\cos x}$$  My multiple attempts are as follows:-  Attempt $1$: $$\int \dfrac{\sqrt{1-2\sin^2x\cos^2x}\cdot\sin x}{\sin^4x\cos x}dx$$  $$\cos x=t$$ $$-\sin x=\dfrac{dt}{dx}$$  $$\int -\dfrac{\sqrt{1-2t^2(1-t^2)}}{t(1-t^2)^2}dt$$ $$\int -\dfrac{\sqrt{2t^4+1-2t^2}}{t(1+t^4-2t^2)}dt$$  $$\int -\dfrac{\sqrt{2t^2+\dfrac{1}{t^2}-2}}{1+t^4-2t^2}dt$$ $$\int -\dfrac{\sqrt{2t^2+\dfrac{1}{t^2}-2}}{\left(\dfrac{1}{t^2}+t^2-2\right)t^2}dt$$ $$\int -\dfrac{\sqrt{\dfrac{2}{t^2}+\dfrac{1}{t^6}-\dfrac{2}{t^4}}}{\left(\dfrac{1}{t^2}+t^2-2\right)}dt$$  Not finding the way to proceed from here.  Attempt $2$:  $$\int \dfrac{\sqrt{\tan^4x+1}\cos x}{\sin^3 x}dx$$ $$\int \dfrac{\sqrt{\tan^4x+1}}{\tan^3 x\cos^2x}dx$$ $$\tan x=t$$ $$\int \dfrac{\sqrt{t^4+1}}{t^3}dt$$  $$t=\sqrt{\tan\theta}$$  $$\dfrac{dt}{d\theta}=\dfrac{1}{2\sqrt{\tan\theta}}\sec^2\theta$$  $$\int \dfrac{\sec\theta\sec^2\theta}{2\tan^2\theta}d\theta$$ $$\int \dfrac{\sec\theta}{2\sin^2\theta}d\theta$$ $$\int \dfrac{\cos\theta}{2\sin^2\theta\cos^2\theta}d\theta$$ $$y=\sin\theta$$ $$\int \dfrac{dy}{2y^2(1-y^2)}$$ $$-\dfrac{1}{2}\cdot\int\dfrac{y^2-(y^2-1)}{y^2(y^2-1)}dy$$  $$-\dfrac{1}{2}\cdot\left(\dfrac{1}{2}\cdot\ln\left|\dfrac{y-1}{y+1}\right|+\dfrac{1}{y}\right)+C$$  $$-\dfrac{1}{4}\cdot\ln\left|\dfrac{\sin\theta-1}{\sin\theta+1}\right|-\dfrac{1}{2\sin\theta}+C$$  $$-\dfrac{1}{4}\cdot\ln\left|\dfrac{t^2-\sqrt{1+t^4}}{t^2+\sqrt{1+t^4}}\right|-\dfrac{\sqrt{1+t^4}}{2t^2}+C$$  $$-\dfrac{1}{4}\cdot\ln\left|\dfrac{\tan^2x-\sqrt{1+\tan^4x}}{\tan^2x+\sqrt{1+\tan^4x}}\right|-\dfrac{\sqrt{1+\tan^4x}}{2\tan^2x}+C$$  $$-\dfrac{1}{4}\cdot\ln\left|\dfrac{1-\sqrt{\cot^4x+1}}{1+\sqrt{\cot^4x+1}}\right|-\dfrac{\sqrt{1+\cot^4x}}{2}+C$$  But real answer is $-\dfrac{\mathrm{cosec}x}{2}-\dfrac{1}{4}\cdot\ln\left|\dfrac{\mathrm{cosec}x-1}{\mathrm{cosec}x+1}\right|+C$  What am I missing here?
Evaluate the following integral:  $$\int \frac{\sqrt{\sin ^4x+\cos ^4x}}{\sin ^3x \cos x }dx$$ where $x \in \big(0,\frac{\pi}{2} \big)$  Could some give me hint as how to approach this question?  I tried to use the fact that $\sin ^4x+\cos ^4x=1-\frac{sin^22x}{2}$ but it didn't help. How should I proceed?
Is "of" instead of "have" correct? Given:  DateTime.UtcNow   How do I get a string which represents the same value in an -compliant format?  Note that ISO 8601 defines a number of similar formats. The specific format I am looking for is:  yyyy-MM-ddTHH:mm:ssZ How do I calculate the run rate in cricket? Will stationary charge radiate E.M waves if observer is accelerating? Why Objective-C,PHP do not support method overloading? Minimal polynomials and cyclic subspaces I'm trying to make my way through two problems in Curtis's Linear Algebra, chapter 25.  One of the two problems is this one, #5:     Prove that $V$ is cyclic relative to a linear transformation $T \in \mathcal{L}(V)$ if and only if the minimal polynomial of $T$ is equal to the characteristic polynomial.   Part of the issue is that I'm not really sure what it's asking. The statement that $V$ is cyclic seems to suggest that the entire vector space (say it has dimension $n$) is generated by some vector $v \in V$, along with $Tv, T^2v, \ldots T^{n-1}v$; Curtis's definition of a cyclic subspace (as opposed to a space, though I can't see why there should be a difference) seems to reinforce this:     A $T$-invariant subspace $V_1$ of $V$ is called cyclic relative to $T$ if $V_1 \neq 0$ and there exists a vector $v_1 \in V_1$ such that $V_1$ is generated by $\{ v_1, Tv_1, T^2v_1, \ldots T^kv_1 \}$ for some $k$.   However, I've seen several examples that seem to directly contradict this interpretation. For example, we recently had a take-home midterm that required we find elementary divisors, rational canonical form, etc., of a $6\times 6$ matrix; a classmate who received a perfect score on her test found that her matrix's characteristic polynomial was equal to its minimal polynomial &#151; it was $(x+8)^2 (x+14)^2 (x^2+61x+80)$ &#151; and yet the vector space could be written as $\left&lt; v_6 \right&gt; \oplus \left&lt; v_4 \right&gt; \oplus \left&lt; -v_2+v_3+v_6 \right&gt;$ &#151; i.e., as the direct sum of three cyclic subspaces, not as one cyclic subspace.  So that suggests that "$V$ is cyclic" might mean the direct sum of cyclic subspaces. However, while working on the exercise that follows (#6), I came across this:  $$S= \left(\begin{array}{ccc} -1 &amp; 0 &amp; 0 &amp; 0\\ 0 &amp; 1 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; 1 &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; 2 \end{array}\right)$$  Here, the characteristic polynomial is not equal to the minimal polynomial, which is $m(x) = (x-1)(x+1)(x+2)$. The eigenspace associated with $\lambda = 1$ has dimension $2$. And so it seems as though, if $V$ has basis $\{ v_1, v_2, v_3, v_4 \}$, we can write $V = \left&lt; v_1 \right&gt; \oplus \left&lt; v_2 \right&gt; \oplus \left&lt; v_3 \right&gt; \oplus \left&lt; v_4 \right&gt; $. This would seem to contradict my second interpretation of the original question.  Obviously I'm missing something. Could someone fill me in on what it might be? (I've tried to be thorough, but I'm also trying not to make this ridiculously long; if I've left out necessary information, I'll be happy to append it.) Creating additional Virtual Host Ubuntu Server Focus existing terminal with `Ctrl-Alt-T` shortcut I tend to use the terminal a lot,  So I wondering if there is a way I can make Ctrl+Alt+T focus the existing terminal if there is one, otherwise create a new terminal? Generating random results by weight in PHP? I know how to generate a random number in PHP but lets say I want a random number between 1-10 but I want more 3,4,5's then 8,9,10's. How is this possible? I would post what I have tried but honestly, I don't even know where to start. Should an adverb go before or after a verb? For example:     The word rarely turns up outside of those contexts.      The word turns up rarely outside of those contexts.   Which one is correct and why? How do I use filenames that start with a dash "-" as command arguments? This may be a simple question, but how do I use files starting with "-" with command line programs?   For example,  I am trying to use pdfgrep on a file named -2013-01-01.pdf but it complains that there are no options defined for 2, 0, 1, 3 and so on.  Changing the filename to something that does not start with "-" solves it, but this is not an option since the files aren't created by me. I simply want to check for a specific change I know is coming. How can Linux/Ubuntu be installed safely on ChromeBook? Is installing Ubuntu on my Chromebook potentially dangerous to anything on my Chromebook? I got it as a gift, so I don't want to destroy anything that is important on the laptop. Also, which is better for being able to safely install Ubuntu: ChrUbuntu or Crouton? I'm using the Samsung Chromebook 3 (as described by the quick-start manual). Does that make any other differences as well? The Quotation Mark character in TeXworks editor has vanished Now, my version of TeXworks is 0.4.5r.1280 and I can't have the " character on the keyboard.    I have ``  or '' instead. Why?  And It's the same with Alt+034. I'm very disappointed with that. Adding materials to more than one object Is it possible to add a texture or material to more than one object at a time? Intuitive explanation of unit root How would you explain intuitively what is a unit root, in the context of the unit root test?   I'm thinking in ways of explaining much like I've founded in .  The case with unit root is that I know (little, by the way) that the unit root test is used to test for stationarity in a time series, but it's just it.   How would you go to explain it to the layperson, or to a person who has studied a very basic probability and statistics course?  UPDATE  I accepted whuber's answer as it is what most reflect what I asked here. But I urge everybody that came here to read Patrick's and Michael's answers also, as they are the natural "next step" in understanding the Unit Root. They use mathematics, but in a very intuitive way. Is there a way to see the history of questions protected by me or another user? Having a record of all actions taken on a site, as is provided by the &quot;activity&quot; and &quot;votes&quot; tabs of our profile, comes in handy very often. Perhaps the most common case is being able to find a question I've interacted with in the past, to see if anything has changed about it. This applies to questions a user has protected as much as anything else. Users are theoretically encouraged to unprotect questions when it's no longer necessary, as emphasized , but in practice they often do not. In many cases it's hard to blame them: unless they specifically marked it as a favourite, the site doesn't keep any record of the question to allow them to find it again. We should help users more easily review their own protections in the future.    For personal review, the votes tab is adequate, but there may also be value in displaying this publicly in the activity tab. There may be users with a history of repeatedly protecting questions over-eagerly. It would be useful to be able to review other users' protection histories to see if this is the case. Even if this is not available to regular users, it may be helpful to moderators, unless their user activity views already allow them to filter down to see this. How to hide TabPage from TabControl How to hide TabPage from TabControl in WinForms 2.0? How is the amplitude of light related to number of photons? If you increase the number of photons leaving a light source you increase the brightness/intensity, so it could be said that intensity is directly proportional to number of photons. However intensity is directly proportional to amplitude squared, so this leads me to wonder that if you increase the amplitude of a light wave, are you just creating more photons rather than affecting each individual photon in some way? Headers of \chapters* take the headers of their previous \chapter When using book.cls and using \chapter* after a \chapter, the header used in \chapter* is wrong. An example is pasted below.   Is there any known workaround for that? I have tried quite some tricks without success.   \documentclass[]{book}  \begin{document}  \chapter{First Chapter} aslkjdf  \chapter*{Unnumbered Chapter} The GNU General Public License is a free, copyleft license for software and other kinds of works.  The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program--to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.  When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things.  To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others.  For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.  Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.  For the developers' and authors' protection, the GPL clearly explains that there is no warranty for this free software. For both users' and authors' sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions.  Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users' freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.  Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.  The precise terms and conditions for copying, distribution and modification follow.  \end{document} Find a bijection from $\mathbb R \to  \mathbb R-\mathbb N$. I want to set my function up such that all natural numbers get mapped to $n+.1$ and reals of the form $n+.1$ to $2n+.1$ is this correct?