How to integrate $\int_{-\infty}^{+\infty} e^{-x^2}\cos x \, dx$
Gaussian-like integral : $\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$
Err 80 appears when I turn on my Canon EOS 5D Mark II Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$ I was solving a few problems from a textbook and I came across this one:     Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through   the point $P=(12,3)$   I could find the tangency points $$ but my solution was completely messy (I had four equations that I had to go back-and-forth with.  Is there an elegant/simple solution to that?  Tangency points:  $$P_1 = (0,3)$$ $$P_2 = \left( \frac{24}{5} , 3-\frac{24}{5} \right)$$ Word style "keep with next" except on last item From the Immediate Window in Visual Studio:   &gt; Path.Combine(@"C:\x", "y") "C:\\x\\y" &gt; Path.Combine(@"C:\x", @"\y") "\\y"   It seems that they should both be the same.    The old FileSystemObject.BuildPath() didn't work this way... How to evaluate the integral $\int \sqrt{1+\sin(x)} dx$ To find:  $$\int \sqrt{1+\sin(x)} dx$$  What I  tried:  I put $\tan(\frac{x}{2}) = t$, using which I got it to:  $$I = 2\int \dfrac{1+t}{(1+t^2)^{\frac{3}{2}}}dt$$  Now I am badly stuck. There seems no way to approach this one. Please give a hint. Also, can we initially to some manipulations on the original integral to make it easy? Thank you. I recentely tried to log-in / sign-up for  using my SE account (which works on ,  and sign-up here) but it showed me an OpenID related error. (See below)  I could reproduce this error across multiple days and browser refreshes and using my other OpenID provider.  Error message (shown in red, obfuscated my e-mail address):  Unable to log in with your OpenID provider:  The openid.return_to parameter (http://area51.stackexchange.com/users/authenticate/?s=bd394805-cf55-4e8e-ad73-21485a09a989&amp;dnoa.userSuppliedIdentifier=https%3A%2F%2Fopenid.stackexchange.com%2Fuser%2F642b1038-5c91-40d4-83ab-eb92db46c3d1) does not match the actual URL (https://area51.stackexchange.com/users/authenticate/?s=bd394805-cf55-4e8e-ad73-21485a09a989&amp;dnoa.userSuppliedIdentifier=https%3A%2F%2Fopenid.stackexchange.com%2Fuser%2F642b1038-5c91-40d4-83ab-eb92db46c3d1&amp;openid.claimed_id=https%3A%2F%2Fopenid.stackexchange.com%2Fuser%2F642b1038-5c91-40d4-83ab-eb92db46c3d1&amp;openid.identity=https%3A%2F%2Fopenid.stackexchange.com%2Fuser%2F642b1038-5c91-40d4-83ab-eb92db46c3d1&amp;openid.sig=4emyzuA31ChHU0LabPGJXldbN8GeKN%2FFBJXJy69yD10%3D&amp;openid.signed=claimed_id%2Cidentity%2Cassoc_handle%2Cop_endpoint%2Creturn_to%2Cresponse_nonce%2Cns.alias3%2Calias3.mode%2Calias3.type.alias1%2Calias3.value.alias1%2Calias3.type.alias2%2Calias3.value.alias2&amp;openid.assoc_handle=hrQL%21IAAAAGzBtFeNGBLU2o04XSWr7ISRAbO-B2tZCK3L61IzAsuQQQAAAAEdoTN6rf9fqaP1oekJGXGXJeB8fNSVuByptp5yx1VpQbAzQwF6HJzh6a0DwedydNJ-wwpJQoN5dOLEFmndZt-L&amp;openid.op_endpoint=https%3A%2F%2Fopenid.stackexchange.com%2Fopenid%2Fprovider&amp;openid.return_to=http%3A%2F%2Farea51.stackexchange.com%2Fusers%2Fauthenticate%2F%3Fs%3Dbd394805-cf55-4e8e-ad73-21485a09a989%26dnoa.userSuppliedIdentifier%3Dhttps%253A%252F%252Fopenid.stackexchange.com%252Fuser%252F642b1038-5c91-40d4-83ab-eb92db46c3d1&amp;openid.response_nonce=2015-06-12T12%3A30%3A42Z7tADDeYZ&amp;openid.mode=id_res&amp;openid.ns=http%3A%2F%2Fspecs.openid.net%2Fauth%2F2.0&amp;openid.ns.alias3=http%3A%2F%2Fopenid.net%2Fsrv%2Fax%2F1.0&amp;openid.alias3.mode=fetch_response&amp;openid.alias3.type.alias1=http%3A%2F%2Faxschema.org%2Fcontact%2Femail&amp;openid.alias3.value.alias1=mycorrectaddress%40web.de&amp;openid.alias3.type.alias2=http%3A%2F%2Faxschema.org%2FnamePerson&amp;openid.alias3.value.alias2=SOJPM) the request was made with.    As noted in the comments I'm not the only one, so I'd guess it may be not something I can fix. So will this be fixed?  Repro steps (using SE OpenID):   Get an Stackexchange OpenID account (like I have) (optional?) Add other OpenID providers and check they're working (StartSSL in my case) Please note: "You" don't have an "account" for the Area 51 subpage yet and still need to do the initial sign-in. Go to the Area 51 SE page Hit "Log-in" in the top-bar Hit the "log-in using stackexchange" button Enter your e-mail and password and submit Get the above error   Repro steps (using other OpenID)   Perform as above (until step 4), but this time step 2 is mandatory Hit "more OpenID options" Provide your OpenID link XXX.provider.com (works on crypto, server fault, ...) Authenticate against your provider Receive a variation of the above error (can be included if needed) Don't show pending suggested edits to me after I run out of votes Who can knight you so that the knighting 'sticks'? Motivating scenario:  I'm a swell upstanding guy, so Ser/Lord X, an acquaintance of mine, knights me. So I walk into my village tavern; the barkeep asks "What'll it be, friend?" and I tell him "That's 'Ser' to you now. I have been knighted by X!" He sneers and goes "well then, X can call you Ser if he likes."  Actual Question:  What about a person's being knighted makes the knighting 'stick'? Is it the identity/standing of the knighter (is that a word?) ? Is it the fact that many other knights walk around telling people that the new guy is also a Knight? Is it the public nature of the knighting?  The  says  about knighting:     This usually happens when a squire reaches adulthood and his master   judges him worthy of accepting the responsibilities of a knight. A man   who has not been raised in the knightly tradition can also be made a   knight as a reward for service. This is often granted to soldiers or   other smallfolk who have shown bravery or performed a great feat.   Notes:   I'm only asking about Westerosi Andal-culture Knights, around the time of Robert I Baratheon's rule. I understand that the word “precious” in the following quote of Maureen Dowd’s article “Heart of Darkness” (NYT, March 20) literally means “2. valuable or important and not to be wasted” as defined in OALD:     Congressman Jones read an e-mail from a former boss of General   Allen’s, giving the congressman this unvarnished assessment:   “Attempting to find a true military and political answer to the   problems in Afghanistan would take decades. Would drain our nation of   precious resources, with the most precious being our sons and daughters. Simply put, the United States cannot solve the Afghan   problem, no matter how brave and determined our troops are.   But I didn’t know “precious” has another meaning as defined in OALD: 5. (disapproving, people and their behavior) very formal, exaggerated and not natural, and Readers English Dictionary: 2.(colloquial) terrible, almost worthless, until I was told by my respected forum mate a few days ago.   Being encouraged by his suggestion for me to “ask another question if I’m not sure of what he means by describing his input as “precious” is actually rather “precious” phrasing, I would like to ask:   What is an example of “precious” being used as “disapproving” or sarcastic way (as neither OALD nor CALD provides examples)? Is there an easy way to discern the instance “precious” being used as “disapproving” from the case being used in the sense of genuine “great value” as used in the above quote other than judging from the context? For a project into balloon simulation I'd like to know how the force on a balloon changes with altitude:  I know that the Buoyant force on a balloon is:  $F = (\rho_{air} - \rho_{helium})gV$  Using changing air density from the international standard atmosphere and changes in gravitational acceleration I can calculate the buoyant force and add this to the (negative) force of gravity mg.  However I know that it isn't this simple, the balloon changes volume with altitude as the air pressure outside the balloon changes.  I'd like to know how the balloon inflates - this would be a great help!  I think it may be something to do with ideal gas laws -  I can calculate air temperature and pressure from the international standard atmosphere but do I use pressure as the air pressure? - or air pressure minus balloon pressure? Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$. Is there any difference in proving "for any integer $n &gt; 0$, $3|(5^{2n}-1)$" and proving "for any integer $n \ge 0$, $3|(5^{2n} -1)$"? Or are they the same? Try to do something vs. Try doing something Is there any differences between      Try to do   And     Try doing   Thank you in advence. Global and local variables in R Viruses: Adaptation to a new host through repeated host jumps A friend told me, during a 3 minute discussion, that viruses that are endemic in host $A$ and make repeated jumps to host $B$ but can't be transmitted between individuals of species $B$, may slowly adapt (through these repeated jumps) to be able to be transmitted between individuals of host $B$ and become epidemic.  I don't know much about epidemiology. I don't understand how a virus population that is endemic to host $A$ may adapt to host $B$ with repeated jumps while the viruses that jump to host $B$ are dead end because they cannot be transmitted further more. Or Are these viruses able to jump back to host $A$ to bring back their newly acquired adaptations to host $B$?  Also, I might misunderstand the meaning of "repeated host jumps transmission". I first thought it meant repeated jumps from a reservoir population in host $A$ to host $B$, but it is also possible that it describes the dynamic of a virus population that is adapted to jump from species to species and they actually gain this ability by keeping jumping and jumping. But then, how could a virus species get adapted to jump from species to species? I'm a bit confused…. Can you give me some hints about this process of cross-species transmission through repeated jumps? What is recordSetVar and where is it used? I was looking at some code and in the 'apex:page' tag, an attribute called recordSetVar was mentioned. I'm new to VisualForce so could someone please tell me when and why is it used? for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|&lt;\infty$ for each $f\in X$, we need to show that there is $M&lt;\infty$ such that  $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$  please tell me how to solve this one, I have no clue. Minimum Length Regular Expression the and thee (I prefer to pronounce it as thuh all the time) My question is can I always pronounce THE with thuh instead of thee? Because unlike "a" "an" rule, pronouncing "thee" seems cumbersome for some people (including me) Note that I know the "emphasis" and "vowel, consonant" rule about THE but I still prefer to pronounce it as thuh everytime. So can I?? What would you consider to be the best exception type to throw when an extension method is called on a null instance (where the extension method does not allow it)? Since extension methods are nothing but static methods you could argue that it should be ArgumentNullException, but on the other hand they're used like instance methods so it might be more natural to use the NullReferenceException. Let's take the following example:  public static string ToInvariantString(this IFormattable value, string format) {     return value.ToString(format, CultureInfo.InvariantCulture); }   This way a NullReferenceException will be thrown if the value parameter is null.   The other example would be:  public static string ToInvariantString(this IFormattable value, string format) {     if (value == null) throw new ArgumentNullException("value");     return value.ToString(format, CultureInfo.InvariantCulture); }   EDIT: In some of the answers you have pointed out that an extension methods can be called like a static method and in those cases a null reference exception would be wrong, which is a great point, and actually one of my concerns, not sure why I forgot to mention that in the question in the first place.   Someone also pointed out that it's wrong to throw a NullReferenceException, and yes, it is. That's why I don't throw it, I just let it happen (let the CLR throw it) by not guarding the method.  I think I favor the ArgumentNullException (that's what I've use so far) but I still think there is at least room to argue for an against the NullReferenceException since it seems more natural in most places where the method is going to be used. Is a black hole singularity a single point? General relativity is expressed in terms of differential geometry, which allows you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the  has whole lines at the top and the bottom that correspond to the North and South poles, respectively; or a single coordinate may refer to multiple points, for example by using inverse geometry the origin refers to all the points infinitely far away.  So, is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate?