Is there a "first" pureblood in the Harry Potter Universe?
Where did the "Magic Blood" of the Harry Potter Universe Originate From?
Integral of $\sin x \cos x$ using two methods differs by a constant? $$ \int \sin \theta\cos \theta~d \theta= \int \frac {1} 2  \sin 2\theta~ d \theta=-\frac {1} 4  \cos 2\theta$$ But, if I let  $$ u=\sin \theta , \text{ then }du=\cos \theta~d\theta $$ Then $$ \int \sin \theta\cos \theta~d \theta= \int u ~ du =\frac { u^2 } 2 =\frac {1} 2 \sin^2 \theta $$ Since $$ \sin^2 \theta =\frac {1} 2 - \frac {1} 2 \cos 2\theta$$ The above can be written as $$ \int \sin \theta\cos \theta~d \theta= \frac {1} 2 \sin^2 \theta =\frac {1} 2 \left( \frac {1} 2 - \frac {1} 2 \cos 2\theta \right)=\frac {1} 4-\frac {1} 4 \cos 2\theta   $$ Why are the two results differ by the constant $1/4$? Thank you. References to a comment on a deleted answer can break Stack Exchange JS For the following URL, JavaScript breaks:  https://stackoverflow.com/questions/16356382/splitting-a-braces-grouped-string-in-python/16356421?noredirect=1#comment23433042_16356421   This is a link to a , the answer I can see as a 10K+ user (although users &lt;10k ).  With Chrome (26, Mac OS X) the console sez:  Uncaught TypeError: Cannot call method 'replace' of undefined   for line 83 of full.js.  Firefox (20, Mac still), the web console spits out:  [12:48:28.459] TypeError: c.attr(...) is undefined @ http://cdn.sstatic.net/js/full.js?v=c9131e40e3fb:83   Safari (6.0.4 on, you guessed it, Mac OS X) has this to complain about:  TypeError: 'undefined' is not an object (evaluating 'c.attr("id").replace')   for line 83 of full.js.  Because of this, the Answer text box on Stack Overflow is lacking all niceties such as an actual WMD editor.  If you remove the hash reference #comment23433042_16356421 from the end of the URL, things work again. Removing the ?noredirect=1 query parameter makes no difference. OL3 How to select features of a WMS layer A word/phrase meaning the "house where I was born"? Make the chat link one click away in the new top bar Why "themselves" and "himself" Can you determine a formula for this problem? Why does a Ubuntu guest in VirtualBox run very, very slowly? I'm using currently Ubuntu 12.04 LTS as a host machine, and I've virtualized three machines:   a Xubuntu 12.04 guest a Windows 7 guest a Ubuntu 12.10 guest   All the guests have the same VirtualBox default settings, but Ubuntu 12.10 is slower!  The installation took 1h 40 min! (Normally, it takes 40 minutes.) Now it is impossible to use it, and all operations are very, very slow.  I'm using VirtualBox version 4.2.4.  Any ideas?  Why is it slow? And how can I make it faster? How to determine function name from inside a function Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ I've been looking at  $$\int\limits_0^\infty  {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$  It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:  $$\displaystyle \int\limits_0^\infty  {\frac{{{x^1}}}{{1 + {x^3}}}dx = } \frac{\pi }{3}\frac{1}{{\sin \frac{\pi }{3}}} = \frac{{2\pi }}{{3\sqrt 3 }}$$  $$\int\limits_0^\infty  {\frac{{{x^1}}}{{1 + {x^4}}}dx = } \frac{\pi }{4}$$  $$\int\limits_0^\infty  {\frac{{{x^2}}}{{1 + {x^5}}}dx = } \frac{\pi }{5}\frac{1}{{\sin \frac{{2\pi }}{5}}}$$  So I guess there must be a closed form - the use of $\Gamma(x)\Gamma(1-x)$ first comess to my mind because of the $\dfrac{{\pi x}}{{\sin \pi x}}$ appearing. Note that the arguments are always the ratio of the exponents, like $\dfrac{1}{4}$, $\dfrac{1}{3}$ and $\dfrac{2}{5}$. Is there any way of finding it? I'll work on it and update with any ideas.    UPDATE:  The integral reduces to finding   $$\int\limits_{ - \infty }^\infty  {\frac{{{e^{a t}}}}{{{e^t} + 1}}dt} $$  With $a =\dfrac{n+1}{m}$ which converges only if   $$0 &lt; a &lt; 1$$  Using series I find the solution is     $$\sum\limits_{k =  - \infty }^\infty  {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}}} $$ In SQL Server 2005, are there any disadvantages to making all character fields nvarchar(MAX) rather than specifying a length explicitly, e.g. nvarchar(255)? (Apart from the obvious one that you aren't able to limit the field length at the database level) Can you please guide me best way to retrieve all active picklist values of all recordtypes of object. Can it be possible through API or any Apex Script.  I want something like below and i will write them in my file:  Obect Name, RecordTypeName, PicklistValues   Thanks Abelian quotient group I'm stuck on the following practice problem.   Any hints would be appreciated.  Suppose $N$ is a normal subgroup of  $G$ such that every subgroup of $N$ is normal in $G$ and $C_{G}(N) \subset N$. Prove that $G/N$ is abelian.  I'm not sure how to use the fact that $C_{G}(N) \subset N$.  Thanks style=apa causes bibliography to fail Is a contradiction enough to prove a set equality to $\varnothing$? Why Didn't People Think That Voldemort Died After His Attack on Baby Harry? After the Avada Kedavra curse rebounded from Harry onto Voldemort, Voldemort disappeared. I'm curious as to why so many key players believed Voldemort was still "alive" somewhere, his powers decimated, but, under the right circumstances, able to return to full power. It seems the witches and wizards in chapter one of Philosopher's Stone, The Boy Who Lived, were happy to believe that Voldemort was gone forever. Why didn't everyone believe Voldemort had died, rather than just disappeared? Was it the lack of a body? Or were there other reasons?  Here's a couple of quotes from Philosopher's Stone if anyone wants to review them. Otherwise, just disregard.     ‘A fine thing it would be if, on the very day You- Know-Who seems to have disappeared at last, the Muggles found out about us all. I suppose he really has gone, Dumbledore?’   ‘It certainly seems so,’ said Dumbledore. ‘We have much to be thankful for.’   Philosopher's Stone - page 13 - Bloomsbury - chapter 1, The Boy Who Lived   ‘But what happened to Vol– sorry – I mean, You-Know-Who?’   ‘Good question, Harry. Disappeared. Vanished. Same night he tried ter kill you. Makes yeh even more famous. That’s the biggest myst’ry, see ... he was gettin’ more an’ more powerful – why’d he go?   ‘Some say he died. Codswallop, in my opinion. Dunno if he had enough human left in him to die. Some say he’s still out there, bidin’ his time, like, but I don’ believe it. People who was on his side came back ter ours. Some of ’em came outta kinda trances. Don’ reckon they could’ve done if he was comin’ back.   ‘Most of us reckon he’s still out there somewhere but lost his powers. Too weak to carry on. ’Cause somethin’ about you finished him, Harry. There was somethin’ goin’ on that night he   hadn’t counted on – I dunno what it was, no one does – but somethin’ about you stumped him, all right.’   Philosopher's Stone - pages 46-47 - Bloomsbury - chapter 4, The Keeper of the Keys Could someone explain the muon $g-2$ experiment problem? Generally accepted root mean square (RMS) error for rectifying topographic maps This is a generic inquiry on the common practice/standard for determining the acceptable values of RMS when georeferencing topographic maps.  Is there an absolute value?  Some literature suggests that it should be "less than or equal to 1/2 of the side of a cell which make up the total resolution of the image." Array#each vs. Array#map Python/Matplotlib - Is there a way to make a discontinuous axis?