I already solved a few integrals with substitution but in this case I have no idea how to start. How to solve the integral $$\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx,$$where $x=\frac{\pi}{2}-t$ with substitution, can you tell me how to start?  It would be great!
Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$   How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?     Is there a simple way to calculate $$ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $$ for $n&gt;3$? Could anyone come up with a reference for this exercise?
Doubt related to a theorem in congruences of Chapter 5 of Apostol Introduction to analytic number theory I am self studying number theory from Introduction to Analytic number theory by Apostol.   I have a doubt in one argument of proof of Theorem 5.14 and I am posting its image.      My doubt is in 2nd line of 2 nd paragraph how Apostol writes - If y is a solution of (5) then ay $\equiv $ at ( mod m) . How does y being a solution of (5) implies that?       Can someone please help lsusb relevant output Bus 001 Device 006: ID 0bda:8812 Realtek Semiconductor Corp. RTL8812AU 802.11a/b/g/n/ac WLAN Adapter  Upon plugging the device it is recognized by lsusb, however when I go to wireless settings no adapter is recognized. The driver is installed (by going to software and updates -> addition drivers and selecting the open source driver). How do you extract IP addresses from files using a regex in a linux shell? Do Israeli security agents interview all those flying to Israel or is it restricted to El Al flights? The one thing I hate about flying to Israel are the security checks at the origin airports. I don't mind X-rays and deep luggage searches, but there's also an annoying interview you have to "pass". The security folks can't really turn you down (especially since I'm a citizen), but they always make it a pain in the butt, because for some reason I fit their profile of a suspicious person.  Can I avoid the personal interviews if I book on a non-Israeli airline? Or perhaps a low-cost airline? How does the wording "for each" work? Some cards in magic has something happen for each {something}. For instance:  makes you gain 2 life for each creature you control. Tapping  adds one green mana to your mana pool, and you gain 1 life for each non-land card revealed.  How is this life gained: all at once or "one bit" at a time? Say I tap Selvala and 1 land, and 2 non-lands are revealed: do I gain 2 life, or do I gain 1 life two times? Would  have one or two +1/+1 counters placed on it if I had it in play at that point? If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality I am trying to prove: If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality.  Set definitions here are from baby Rudin.   Attempt: There are 4 possible cases:  1.A is countable and B is finite.  2.A is countable and B is countable.  3.A is uncountable and B is finite.  4.A is uncountable and B is countable.  Then for each case I need to show that $A \cup B \sim A$. I can prove 1-3, but I am unsure how to prove case 4. And in general, it seems that my proof strategy is inefficient and there exist much faster proof then proof by cases. This is a  about DoS and DDoS mitigation.   I found a massive traffic spike on a website that I host today; I am getting thousands of connections a second and I see I'm using all 100Mbps of my available bandwidth.  Nobody can access my site because all the requests time out, and I can't even log into the server because SSH times out too!  This has happened a couple times before, and each time it's lasted a couple hours and gone away on its own.  Occasionally, my website has another distinct but related problem: my server's load average (which is usually around .25) rockets up to 20 or more and nobody can access my site just the same as the other case.  It also goes away after a few hours.  Restarting my server doesn't help; what can I do to make my site accessible again, and what is happening?  Relatedly, I found once that for a day or two, every time I started my service, it got a connection from a particular IP address and then crashed.  As soon as I started it up again, this happened again and it crashed again.  How is that similar, and what can I do about it? How to restart the networking service? I tried to use  sudo service networking restart   and  sudo /etc/init.d/network restart   but they both crash the window manager and I can no longer use my keyboard for input into X.  when I use the /etc/init.d/ method it complains saying that I should use the service utility  e.g. service networking restart   but it crashes just the same.  Is there a GUI method of restarting networking? Create line graph with ggplot2, using time periods as x-variable HTML and CSS is not working? I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \sum\nolimits_{k=1}^n \frac{1}{k} - \log(n)$ converges. At first I want to know if my answer is OK.  My try: $\lim\limits_{n\to\infty} \sum\limits_{k=1}^n \frac{1}{k} - \log (n) = \lim\limits_{n\to\infty} \sum\limits_{k=1}^n \frac{1}{k} + \sum\limits_{k=1}^{n-1} \log(k)-\log(k+1) = \lim\limits_{n\to\infty} \frac{1}{n} + \sum\limits_{k=1}^{n-1} \log(\frac{k}{k+1})+\frac{1}{k}$     $ = \sum\limits_{k=1}^{\infty} \frac{1}{k}-\log(\frac{k+1}{k})$ Now we prove that the last sum converges by the comparison test: $\frac{1}{k}-\log(\frac{k+1}{k}) &lt; \frac{1}{k^2} \Leftrightarrow k&lt;k^2\log(\frac{k+1}{k})+1$ which surely holds for $k\geqslant 1$      As $ \sum\limits_{k=1}^{\infty} \frac{1}{k^2}$ converges $ \Rightarrow \sum\limits_{k=1}^{\infty} \frac{1}{k}-\log(\frac{k+1}{k})$ converges and we name this limit $\gamma$ q.e.d Is there a quick way to get to the Control Panel in Windows 8? Obviously in all previous versions of Windows you could simply click START and then select "Control Panel" to get to it, but on Windows 8 there is no such option.  Is there an easy way to quickly get to it without having to press WIN, type "Control Panel", and then press Enter? Book: World covered in trees that are miles high Using "the" instead of "a" for something that isn't a specific object I've been curious about this for a while.  Generally, one uses "the" if they mean a specific object ("The copy of Lord of the Rings I bought yesterday") instead of any object that fits a description ("I have a copy of Lord of the Rings").  However I've seen "the" used in some cases where the phrase doesn't refer to a specific object. I've noticed it mostly being about technology, such as "This game requires the Classic Controller" (seen on some Wii Virtual Console games). While saying "This game requires a Classic Controller" seems to be correct as well, using "the" doesn't feel wrong either.  The controller's page on Wikipedia also uses "the":   What is the rule that allows the use of "the" in this context?  I presume it has something to do with "Classic Controller" being a proper noun. Can't move files to trash without any permission issues This is not like one of the cases addressed before - I've checked them all.  I've suddenly begun getting that message "... can't be put in the trash. Do you want to delete it immediately?" this morning in my Home directory only.  I have a Home partition (ext4, mounted properly) with symlinks for user folders like Downloads, Documents, etc. to folders with the same names in a Data partition (ntfs, mounted properly).  For example, I cannot move a file in Downloads folder to Trash when I try via Home, but no problem if I try via Data partition, although the file in question has the same ownership and permissions at both locations.  Also no problem at all with files in my Home folder, including sub-folders like Public that are not symlinked to my Data partition.  Interesting puzzle for me...  PS: Just upgraded to 15.04 but I don't think it matters... On second thought, it might perhaps be something to do with upgrading from Nautilus 3.10 to 3.14?  PPS: This bug has been finally resolved some time later in version 16.10, and no longer exists under Ubuntu 17.04. $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$ When is double-quoting necessary? The old advice used to be to double-quote any expression involving a $VARIABLE, at least if one wanted it to be interpreted by the shell as one single item, otherwise, any spaces in the content of $VARIABLE would throw off the shell.  I understand, however, that in more recent versions of shells, double-quoting is no longer always needed (at least for the purpose described above).  For instance, in bash:  % FOO='bar baz' % [ $FOO = 'bar baz' ] &amp;&amp; echo OK bash: [: too many arguments % [[ $FOO = 'bar baz' ]] &amp;&amp; echo OK OK % touch 'bar baz' % ls $FOO ls: cannot access bar: No such file or directory ls: cannot access baz: No such file or directory   In zsh, on the other hand, the same three commands succeed.  Therefore, based on this experiment, it seems that, in bash, one can omit the double quotes inside [[ ... ]], but not inside [ ... ] nor in command-line arguments, whereas, in zsh, the double quotes may be omitted in all these cases.  But inferring general rules from anecdotal examples like the above is a chancy proposition.  It would be nice to see a summary of when double-quoting is necessary.  I'm primarily interested in zsh, bash, and /bin/sh. Please stop using SoftwareEngineering.SE as your toilet bowl Stack Overflow users have been using the "off topic - belongs on SoftwareEngineering.SE" close reason as an alternative to all the other close reasons. It'd be helpful if people actually read the FAQ and the six  before voting to migrate questions like:   , no redeeming value to favorite lists that become argumentative , "help my exact situation" questions are too localized , fails 5 out of the 6 guidelines , fails 4 out of 6 , too localized and too vague , too broad   Absent a reversal of the Stack Overflow community from viewing SoftwareEngineering.SE as the Stack Overflow cesspool, can the "off topic - belongs on SoftwareEngineering.SE" close reason be removed in favor of manual migration by moderators who can actually spend the 10 seconds to determine if the question should be closed outright first?    To say that SoftwareEngineering.SE is such a nebulous place and nobody knows what its scope is as the reason why crappy questions keep getting migrated there is to miss the point. Good questions are good questions on all of the sites, and questions that are too vague, that are too argumentative, too localized, and too broad are universal close reasons. There is no special dispensation for SoftwareEngineering.SE to have bad questions, even if you personally think the questions on there suck.  Even the six subjective guidelines for subjective questions are universal guidelines: they affect all sites. So even if you had no idea what the consequences of using the "off topic - belongs on SoftwareEngineering.SE" close reason were, there are still all of the other close reasons for which bad questions, like the ones above, can be closed. $a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus I noticed relation between modulo operation and number which is power of two  Example I have to calculate $ 3431242341 \mod 2^5 $, which is $ 5 $ but it is equivalent to  $ ( 3431242341 \mod 2^9 ) \mod 2^5 $   I tried many examples and it seems to be true in general, and I am not sure if it is a coincidence or true in general that I can use first modulo operation ( greater number) and the result will be the same. Determine accuracy of model which estimates probability of event