Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $ I know that $$ \int_0^\infty \frac{\log x}{\exp x} = -\gamma $$ where $ \gamma $ is the Euler-Mascheroni constant, but I have no idea how to prove this.  The series definition of $ \gamma $ leads me to believe that I should break the integrand into a series and interchange the summation and integration, but I can't think of a good series. The Maclaurin series of $ \ln x $ isn't applicable as the domain of $ x $ is not correct and I can't seem to manipulate the integrand so that such a Maclaurin series will work.   Another thing I thought of was using $ x \mapsto \log u $ to get $ \int\limits_{-\infty}^\infty \frac{\log \log u}{u^2} \ du $ and use some sort of contour integration, but I can't see how that would work out either.
Integral representation of Euler's constant Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$  where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).    This integral was mentioned in  as in  , but the solutions I've got uses corollaries from . Can you give me a simple solution (not using much advanced theorems) or at least some hints.
Is there a way to export each rendered layer as its own image? My scene uses multiple render layers.  I'd like to edit each rendered layer in Photoshop, and composite it in Photoshop.  Is there a way to export each layer as an individual image? How do I correctly tag my questions? Here are some tips for tagging questions. These guidelines will help you more accurately tag your questions, which in turn will help them get more attention and lead to better answers faster.     For more information, see "" in the . Why is January month 0 in Java Calendar? In java.util.Calendar, January is defined as month 0, not month 1. Is there any specific reason to that ?  I have seen many people getting confused about that... In the book I'm study is written:     A normal subgroup of a group need not be characteristic.   And as an exercise I'm supposed to find an example, it also said that is pretty hard to find one. After trying for two days I wasn't able to find one example.   So, I'm asking for an example of a group $G$ with a normal subgroup $H$ that is not a characteristic of $G$.  I will add some context because maybe it will clarify why the book said it is difficult to find an example: the main problem with the exercise is that until it the book had only covered Group Definition, Subgroups, Lagrange's Theorem and Homomorphisms. So I'm supposed to find a example with such lack of advanced tools. An inequality about Stirling numbers For all positive integers $n,r,k$ that $n=rk$, how to prove the inequality $$(r-1)!^k\le \frac{c(n,k)}{S(n,k)}\le (n-k)!$$ $c(n,k)$ is unsigned Stirling numbers of the first kind and $S(n,k)$ is Stirling numbers of the second kind.  Any help would be appreciated! A gold reviewer badge? Arithmetic of continued fractions, does it exist? I'm interested in the arithmmetic of continued fractions and specially in multiplication. Consider  $$ f(x)=\cfrac{f_{0}(x)}{1-\cfrac{f_{1}(x)}{1+f_{1}(x)-\cfrac{f_{2}(x)}{1+f_{2}(x)-\cfrac{f_{3}(x)}{1+f_{3}(x)-\cdots}}}} $$   and   $$ g(x)=\cfrac{g_{0}(x)}{1-\cfrac{g_{1}(x)}{1+g_{1}(x)-\cfrac{g_{2}(x)}{1+g_{2}(x)-\cfrac{g_{3}(x)}{1+g_{3}(x)-\cdots}}}} $$   Are there arithmetic rules (algorithms) for the multiplication of continued fractions? Specifically, is it possible to obtain a continued fraction $h(x)=f(x)\cdot g(x)$ where   $$ h(x)=\cfrac{h_{0}(x)}{1-\cfrac{h_{1}(x)}{1+h_{1}(x)-\cfrac{h_{2}(x)}{1+h_{2}(x)-\cfrac{h_{3}(x)}{1+h_{3}(x)-\cdots}}}} $$   I've found , but I'd like more. Does anyone here knows of papers, algorithms, etc?  Thanks.    Update 1: Another . More options when flagging for migration Windows 10 April 2018 Update added an extra Language and I cannot remove it I have just updated to Windows 10 April 2018 Update (1803 / 17134.1)  I had my laptop configured completely as UK with only a UK keyboard layout  Since the update, a US keyboard has been added so I now see the switch icon in my task bar    It appears that all the keyboard language settings are now migrated to the new Settings app and no longer in the old Control Panel    However, as can be seen from the screenshot above, the Remove button under the newly added US layout is greyed out (as is the UK one)  How can I remove the US layout?  UPDATE 1: I just updated my desktop and it hasn't added a US keyboard!  UPDATE 2: I added a third language (French) and was able to remove it, the button was enabled. The US remove button remained disabled both after adding French and after removing it.  UPDATE 3: I added another English language (Australian) and this cannot be removed! Button is disabled  UPDATE 4: I followed this  except I removed the values from the keys for all users and also the values under the Substitutions key. The task bar switch icon is gone but I still have all 3 languages in Settings still How come I can't give more than 500 rep? Why can't I give more than 500 reputation as a bounty at a time? The system won't let me choose an option more than 500.  Also, how were some users able to do so? On  the OP has given a single bounty worth 550 rep, but at the moment I have 844 rep and I can't give any more than 500 rep on a bounty.  Can you please tell us if this is a bug or not? Did the verb “fire a weapon” exist before the actual introduction of firearms on battlefields?  More specifically, does it make sense for a creative work to have archers (or whatever ranged weaponry) be told to “fire!”, when the world they live in has not yet seen firearms?  It seems some kind of an anachronism to me, since before firearms, “fire” would never propel any projectile...  I've seen several movies do it; I can't remember them all, but for instance I verified it in The Lord of the Rings trilogy (left), Kingdom of Heaven (top right), and more recently Frozen (bottom right):    The first two do it several times (with Kingdom of Heaven also using it on ballistae, and The Return of the King even having Aragorn ask Legolas to ), whereas firearms never appear on the battlefields they feature (Kingdom of Heaven is set in the 12th century; as for The Lord of the Rings, apart from one occurrence of a witchcraft-ish bomb, it's archery and medieval siege engines all the way).  Frozen may be debatable, happening probably somewhere in the 18th or 19th century, but since we only see swords, spears and the like, it made me flinch to hear crossbowmen use “fire!”.  I've also noticed other movies avoiding this, using “loose!” instead, such as Troy (left) and Gladiator (right):    This makes me think it would be the right thing to say instead... although I'm still wondering why not use “shoot”, which seems simpler to me. Scientific visualization - How to replicate instances around a sphere? I want to replicate a protein structure around an icosahedron. Basically a vesicle of protein structure is what we want:  When I am instancing the protein structure using each vertex of the icosahedron it looks like:  However, the tails are supposed to be directed towards the center. How can I make that work? I'm loading elements via AJAX. Some of them are only visible if you scroll down the page. Is there any way I can know if an element is now in the visible part of the page? Why are processes in Windows 8 32 bit? Should the currently selected item in a combo/dropdown box also be included in the list of choices? For example, you have a combo box which contains 10 elements including the one that is currently selected. When you prompt to drop down the list to choose another, should the currently selected item also be in this list? Why have  on every line Unusual placing of a second impression of a USA visa immigration stamp. Does it mean anything? My parents traveled to the USA a couple of weeks ago (Caracas - Miami). It wasn't their first time; they've probably come and gone a dozen times by now, and they've never had a single issue. This time, the immigration guy was kind of pompous and, once they told him they weren't staying at a hotel but with some friends, he began looking at them somewhat funny. Finally, he stamped their passports (normally, with entry date and a 6-month term of stay) but he put an additional stamp, one that no one they asked had seen before:    It seems to be the same usual entry stamp, but cut in half. What does it mean? Should we be concerned? Duplicate application icons in Ubuntu dock upon launch What can I do to manage my phone's internal storage? Prove : $p$(n│even number of ODD parts)=$p$(n│distinct parts ,number of ODD parts is even )