How to compute the integral $\int_0^1 \frac{x\ln x}{\ln (1-x)}dx$?   We can write  \begin{align*} \int_0^1 \frac{x\ln x}{\ln (1-x)}dx=\int_0^1\frac{(1-u)\ln (1-u)}{\ln u}du =-\int_0^1\frac{1-u}{\ln u}\sum_{n=1}^\infty \frac{u^n}{n}du \end{align*} Seeing \begin{align*} \int_0^1\frac{u^{a-1}-u^{b-1}}{\ln u}du=\ln\frac{a}{b},\forall a,b&gt;0, \end{align*} we get  \begin{align*} \int_0^1 \frac{x\ln x}{\ln (1-x)}dx=\sum_{n=1}^\infty \frac{1}{n}\ln \frac{n+2}{n+1}. \end{align*} I have tried hard to compute the series above ,but without any progress.
Evaluate $$I=\int_{0}^{1}\frac{x\log x}{\log(1-x)}dx$$   I try to change it to:$$I=\int_{0}^{1}\frac{(1-x)\log(1-x)}{\log x}dx$$ but I can't do next steps,help me ,thank you very much.
Does the verb "unvelcro" exist or what can we use instead? I know that the following word pairs exist:  button – unbutton tie – untie  Can we use an analogous process with the verb &quot;velcro&quot;? What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity? I tried and got this   $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an integer. $$\lim_{n\to\infty}n\sin(2\pi en!)=\lim_{n\to\infty}n\sin\left(2\pi n!\sum_{k=0}^n\frac{1}{k!}\right)=\lim_{n\to\infty}n\sin(2\pi m)=\lim_{n\to\infty}n\cdot0=0$$  Is it correct? What is a plain English explanation of "Big O" notation? I'd prefer as little formal definition as possible and simple mathematics. Is it possible to keep publishing under my professional (maiden) name, different from my married legal name? Extract substring in Bash Given a filename in the form someletters_12345_moreleters.ext, I want to extract the 5 digits and put them into a variable.  So to emphasize the point, I have a filename with x number of characters then a five digit sequence surrounded by a single underscore on either side then another set of x number of characters.  I want to take the 5 digit number and put that into a variable.  I am very interested in the number of different ways that this can be accomplished. Bluetooth Broadcom 43142 isn't working Where is the cron / crontab log? How to combine array items with string substitution? Given a function:  function x(arg) { return 30; }   You can call it two ways:  result = x(4); result = new x(4);   The first returns 30, the second returns an object.  How can you detect which way the function was called inside the function itself?  Whatever your solution is, it must work with the following invocation as well:  var Z = new x();  Z.lolol = x;  Z.lolol();   All the solutions currently think the Z.lolol() is calling it as a constructor. Can we create a subsection of the References Cited section with different reference numbering? I need to add some more publications. I do not want to change the previous reference numbers, so I need to create new citations with difference reference numbering like [p1], [p2], etc. Can LaTeX do that? SF novel about one-way scientific expedition to inhabited planet, where telepathic aliens "talk" sort of ++like this++ and ~*~this~*~ I'm looking for a novel I read in the mid-to-late 1990s. It was a science fiction novel about an expedition of scientists sent at sub-light-speed (maybe in cryo-sleep?) to investigate a star system. Once there they set up a permanent scientific outpost.  There was at least one inhabited planet in the system, with a sentient species that communicated telepathically. The aliens' speech was represented with various symbols instead of quotation marks, with each named character having (I think) its own characteristic set of symbols, as in the title of the question. The scientists are eventually able to communicate with these (generally friendly) aliens.  The thing I remember most clearly is the end of the book. This was a one-way trip for the scientists, so gradually they all die off from various causes. When the next-to-last member of the crew dies of old age, the survivor is looking for an epigraph for her funeral, and (ironically, considering my current plight) can't find it in his search of the electronic library. He finally discovers it under "humor" because it's by Mark Twain: "Wherever she was, there was Eden." (This quote, of course, I found immediately in  and many other sources; it's from Eve's Diary.)  Other pieces of information I'm less sure of:   I don't think the planet was hospitable to humans; it may have been something like a gas giant, with the aliens "swimming" through the gas. I think the scientists stayed in an orbital space station, though they may have made expeditions to the planet in vehicles or protective suits. There may have been a "twin" planet, or a moon in an eccentric orbit, that caused periodic "flares" of some sort on the aliens' planet. There might have been some conflict between the crew early on, maybe because there was a non-scientific (military) commander, or perhaps just because people have conflict. I think the author was of sort of middling fame; respected and known by SF enthusiasts, but not a superstar like Asimov or Clarke. Almost certainly male, and probably with a professional science background. I think the science was fairly authentic, although the first-contact bit was obviously highly speculative. What are the disadvantages of using a Zener diode over a linear voltage regulator? I wonder why a Zener diode is worse than linear voltage regulator.  Some of the posts online mentioned that because of the power dissipating problem. Zener diodes keep consuming a large amount of current while the load may just need a very little amount of current.   If I understand correctly, this phenomenon will happen only when the Zener diode is connected with the load in parallel. What if I connected my Zener diode with my load in series? Also, someone said that linear voltage regulators (e.g. LM7805) are not good voltage regulators as the power dissipated in them is much higher than that in switching voltage regulators. In my understanding, the power dissipated in a linear voltage regulator can be calculated by dropout voltage x current = power converted to heat.   Regarding these 2 points, other than a linear voltage regulator can handler a wider current range, what is the benefit of choosing a linear voltage regulator over a Zener diode? Phone keeps rebooting on Volume Up + Power (trying to install ROM) I tried to install Android's ROM and got ERROR 7, then I wiped out cache, data, internal storage, where was twrp.apk installed. After that I cant get into boot menu (with Install, Wipe... buttons). Then I copied twrp to the phone: sudo fastboot flash recovery &lt;twrp.img&gt; Tried versions 3.2.1-0 and 3.4.0-0, because I heard for certain phone (I have Xiaomi Mi4) it's necessary to use proper version, though I don't know which. What's going on now: I can successfully can get to the nice splash screen using Volume Down + Power, then on my PC I give sudo fastboot reboot command and press Volume Up + Power and here is the problem: It has no effect, it just keeps rebooting (I see only Xiaomi's logo). As a grad student, I just find it tremendously frustrating when emails are ignored by faculty (both at my institution and at other places). Is this acceptable? I'd be content with a simple response with one line along the lines of "I have read this email and am busy right now, will respond in X days when I can write a more detailed email." I understand that, as a first year student in my field, I'm not really in a position to request anything of anyone, but it just seems like common decency.  Instead, I never hear back, as if my email has been jettisoned off into oblivion. I notice that in a pdf file compiled from Latex, "fi" such as in "field" cannot be separated as "f" and "i" when copying text out of the pdf file. I wonder why and if this can be changed? Thanks and regards! Is it possible to use a spell scroll with a spell that requires a bonus action to cast? The DMG contains spell scrolls (on page 200), with the following rule:     If the spell is on your class's spell list, you can use an action to read the scroll and cast its spell without having to provide any of the spell's components.   This is all well and good for spells with a casting time of 1 action (which admittedly is most of them), but for spells with a bonus action casting time, how is it supposed to work?  Does it require:   Both an action and a bonus action?  Just an action? Just a bonus action? Something else I haven't thought of?   In case it needs saying, I'm not interested in speculation here - only RAW-based answers or explicit statements of developer intent should apply. Emails Per Invocation Still Capped at 100? Consider a particle on the ground. This particle is raised by a force of magnitude $mg$ to a height $h$ above the ground. At this point, the work done on the particle by the force is $mgh$, which is equal to the potential energy of the particle. But, during this period, the force of gravity also acts on the particle and is displaced by $h$, and so does a work of $-mgh$ on the particle. Shouldn't  the two cancel and no net work should be done on the particle?   If they don't cancel, then where did the energy that came from the work done by the force of gravity go? A word for "mercy killing" "Mercy killing" is an act of killing someone who is already dying and in terrible pain, in order to reduce his or her suffering.    Is there any single word for "mercy killing"? Calculate line distances inside of polygons using ArcGIS for Desktop? I'm working with two data sets:   NHD polyline stream layer for my entire state HUC Watershed boundary Polygons for the entire state   I am looking for a way to calculate the total length of streams within each watershed boundary polygon, and would ideally like to have each polygon then attributed with the sum of stream miles within its boundary.  Can anyone help me?