Find $\displaystyle \int \sqrt{\tan(x)}dx$.   According to an integral calculator, the answer to this question is $$(-2 \tan^{-1}(1-\sqrt(2) \sqrt(\tan(x)))+2 \tan^{-1}(\sqrt(2) \sqrt(\tan(x))+1)+\log(\tan(x)-\sqrt(2) \sqrt(\tan(x))+1)-\log(\tan(x)+\sqrt(2) \sqrt(\tan(x))+1))/(2 \sqrt(2))+C.$$   Is the question as complicated as the answer looks?
I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$
Phrase to give a non-detailed answer/explanation What expression could I use when giving non-detailed answers/explanations about how something works? Let's say I want to explain how a printer works without getting into too much detail, Would it be OK to say:   "At a very superficial level, a printer ...." "At a very high level, a printer ..."   I feel like there are more appropriate expressions to begin such type of sentence. Any help would be greatly appreciated. What is the LaTeX command for "em dash" (—)? I am trying to insert the symbol in the title of the question, which I copied from a PDF article, but it appears as space in the output file. What is the corresponding LaTeX command? Validationengine jquery not working properly for localization I'm trying to make a list of ways to tell if a given degree sequence is impossible. For example $3,1,1$  is not possible because there are only 3 vertices in total so one can't have degree 3.  The list so far   vertices has degree equal to or larger than number of vertices sum of degrees is odd for n vertices if one has degree n-1 and another has degree 0 for n vertices the sum of the degrees cannot be greater than $n(n-1)$ because this would be have more edges than a complete graph Proving that orthogonal vectors are linearly independent I am trying to prove that $a_1$, $a_2$, $a_3$ are linearly independent.  I am asked to use vector product and prove that if $c_{1}a_{1} + c_{2}a_{2} + c_{3}a_{3} = 0$ then $c_1 = c_2 = c_3 = 0$  I am completely stuck on where to go with this problem. I would think that linearly independent then the null space of the space can only be equal to $0$. Would this prove it? bash script – for i in {x..y} i am writing a script in bash and I would like to give the two numbers in the brackets, x and y to be precise, as variables.  I wrote the script this way:  echo "x?" read x echo "y" read y  for i in {$x..$y}   but it does not work.  How could I set it up to make it function?  Thanks a lot in advance. After the performance problems we have run into with  we've decided to make a change, we're moving the network on to .  Here's where to get started:   What works:   All search operators should be in Many changes below from the old search behavior   What's different:   The new search is AND, adding a search term will narrow your results, not expand them. The look of search results is now more unified with the following differences between Q&amp;A:   "Q:" or "A:" prefix Answer counts and tags will be present in questions. The answer count will be highlighted in green if it's accepted. (We may index other things later, that's why the results are more generic/google-y now.)  It's faster, about 5-10x faster for almost all searches on large sites like Stack&nbsp;Overflow Quoted phrases are exact matches except for case-sensitivity, for example,  or . If not in a quoted phrase, words will now be stemmed () Snippets for context will now be more relevant Excludes are now explicitly supported (like -term and -"my phrase") We don't roll up to the question anymore, if you searched for something that matched an answer, we'll show you the answer directly in search results The search per minute limit is effectively gone, it'll block a bot, but no one else should hit it... We'll be monitoring performance though infavorites:mine is back, with infavorites:12345 (any user id) added   What needs work:   What you tell us is broken... Do your worst.   Testing, not yet final:   New range support, here are some examples:   answers:1..1 (exactly one answer) answers:10..20 (10 to 20 answers) created:2008..2009 (created between Jan 1st, 2008 and Dec 31, 2009) created:2010-04..2010-05 (created between April 1st, 2010 and May 31st, 2010) created:2011-01-05..2011-01-06 (to be clear: it goes to end-of-day) This range syntax applies to a new advanced operator: lastactive: and the other numerical ranges, views:, and score:  Additionally, the above syntax is a bit more flexible if you forget it exactly (e.g. score:20-30) in an effort to be intuitive as possible, give it a try - tell us what you think.    I'll try and improve the search as much as possible before it becomes default... At the very least we want it equivalent in relevance and functionality before the switch, but ideally much better. We won't switch until then.  Please give it a try, tell us what you think here, and we'll continually improve it over the holidays as time allows. Bear in mind you may get a search offline page during this test if a change we're making for results necessitates a reindex, and even that should only last about two minutes here on Meta&nbsp;Stack&nbsp;Overflow.  Give it your worst, compare results to the old search, tell us what's better and what's not. We look forward to the feedback.  Update The new search is now the default across the network - we'll be monitoring things over the next few days and would appreciate reports of any crazy behavior you see here. I am developing a set of classes that implement a common interface. A consumer of my library shall expect each of these classes to implement a certain set of static functions. Is there anyway that I can decorate these class so that the compiler will catch the case where one of the functions is not implemented.  I know it will eventually be caught when building the consuming code. And I also know how to get around this problem using a kind of factory class.  Just curious to know if there is any syntax/attributes out there for requiring static functions on a class.  Ed Removed the word 'interface' to avoid confusion. Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)&lt;3$   for all positive integers $n$   This problem is copied from Math Olympiad Treasures by Titu Andreescu and Bogdan Enescu.They start by stating that induction wouldn't directly work here since the right hand side stays constant while the left increases.They get rid of this problem by strengthening the hypothesis.$$\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)\le3-\dfrac{1}{n}$$ and then proceed by induction.  The problem is that I can't find a motivation for the above change.I mean,we could have subtracted a lot of things from the RHS but what should nudge us to try $\dfrac{1}{n}$?The rest of the proof is quite standard,but I can't see how I am supposed to have thought of it.Is it just experience?Or is it a standard technique?A little guidance and motivation will be appreciated. Is something wrong with StackOverflow's spam filter? Recently there have been huge numbers of spam topics related to football streaming. They have all been spam flagged and deleted but there are more getting posted all the time, I just flagged another 6 or so a few minutes ago. This has been going on for around 3 days now with these topics getting constantly posted over and over again in the  tag. Adaline is 100-something years old, and her daughter appears to be 90 years old. Well, Adaline gave birth to her in her mid 20's, and to be precise, she should be 80.   Anyways, Adaline's old lover's son (Ellis) is probably in his late 20's/early 30's. And Adaline has stopped aging at 29, which is pretty close to Ellis. Her old lover is said to be a few years younger than her and still, his son is so young. Also, Adaline could make him believe that she is the daughter of her own old self.   Then,      how can Adaline's real life daughter be so old?   I can't make sense out of this.  Why is it this way? The age pattern of the actors and actresses is inconsistent. It seems to me that the filmmakers have exaggerated some things. The Help Vampire problem Initial note: Having read , the answers there do not help in this specific situation, as they rely on having lots space to spare - I have very little.  When I started my city, I designed a simple exit from the default highway that I thought was sustainable: I left enough space around it to upgrade the road as I needed to, which has worked up until now. However, I am now stuck with a six-lane one-way exit from the highway that isn't getting used: since cars seem to pick the most efficient lane to get where they want to go, regardless of traffic, the rightmost lane fills up, then spills over onto the highway. This blocks the highway, my bus routes, and more importantly my planned emergency routes.  My setup looks like this:    The red arrows show road direction, and the orange line is the traffic backed up. Note that in this image, I have replaced the six-lane one-way exit with a highway exit, as this is just about as efficient...  This situation is getting to be pretty bad, and is bringing half my city to a standstill. Its effects compound themselves further down the highway:    In this image, the orange line is the traffic directly caused by this exit not working. This traffic jam then blocks two exits/entrances further down, on the red crosses; the yellow line shows the traffic jam caused by these being blocked. In other words, my inefficient exit design is now blocking itself from moving.  So this is a two part problem.   Considering that I'm constructing a residential district in the space above left of the problem exit, and would like to keep the solution inside the power line, how can I redesign this exit to be more efficient? How do I solve the ring of traffic jams? Is it a simple case of just deleting and replacing a bit of road? How to use an optimization solver to get t-stats and p-values for the estimates? Which one sounds better? "of mine" or "my" My friend (or "a friend of mine"?) told me that anytime I need to express such expression, it's always better to say "of mine", because it sounds more naturally.  Like "two brothers of mine" rather than "my two brothers".  Is that true? If $\lim_{x\to+\infty}[f(x+1)-f(x)]= \ell,$ then $\lim\limits_{x\to+\infty}\frac{f(x)}x=\ell$. Let $f:[0,+\infty)\to\Bbb R$ be a function bounded on each finite interval. I want to show that if $\lim\limits_{x\to+\infty}[f(x+1)-f(x)]= L,$ then also $\lim\limits_{x\to+\infty}\dfrac{f(x)}x = L$ How do I separate hydraulic disc brake pads after pulling the brake while the wheel is off? Are there uncountably many surjections from $\mathbb N$ to $\mathbb N$? Statement: There are uncountably many surjections from $\mathbb N$ to $\mathbb N$.  Is the following proof valid:  The real numbers are uncountable.   There exists a bijection between the set of real numbers and the set of all functions from $\mathbb N$ to $\{0,1\}$.   Given a function, $f$, from $\mathbb N$ to $\{0,1\}$, construct a surjection from $\mathbb N$ to $\mathbb N$, denoted $g$, as follows:  We map the $n$th composite number to $n$, and the nth prime, to $f(n)$.   Clearly this is a surjection since the composite numbers are countable. Clearly different functions $f$ lead to different surjections $g$. Since there are uncountably many $f$s, there are uncountably many surjections $g$. Is there a reason behind the ordering of letters in the English alphabet? Is there a reason behind the ordering of letters in the English alphabet? i.e. why are we taught “A,B,C,D,E,F,...,Z”? Why not “L,A,S,U,I,Z,...,C”?  I am asking this because, in some of the languages I know, I am told that the ordering of the letters in the alphabet is based on the ease with which they can be pronounced or the frequency with which they are used or depending on the part of the vocal cord that needs to be stressed to pronounce the letter.  Is there a similar rationale?  EDIT  You can see a couple of references for arrangement of letters in Sanskrit  and . Could it be that Goldbach conjecture is undecidable?