$$I = \int_0^1\dfrac{t \ln t}{\sqrt{1-t^2}}\mathrm{dt}$$   Attempt:   Let $t = \sin x$  $\implies dt = \cos x dx$  $\implies I = \displaystyle \int_0^{\pi/2} \dfrac{\sin x \ln (\sin x) \cos x dx}{\cos x}$  $\implies  I = -\ln (\sin x) \cos x|_{0}^{\pi/2}   + \displaystyle\int_0^{\pi/2}\dfrac{\cos^2 x}{\sin x} dx$  The second integral can be "evaluated"  using $\cos^2 x = 1- \sin^2 x$. But unfortunately, there are finally two divergent integrals.   $\implies  I = -\ln (\sin x) \cos x|_{0}^{\pi/2}   + \displaystyle\int_0^{\pi/2}\csc  x dx$ + $\displaystyle\int_0^{\pi/2}\sin x dx$  How do I handle this?
Evaluation of $\displaystyle \int_{0}^{1}\frac{x\ln (x)}{\sqrt{1-x^2}}dx$   $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{1}\frac{x\ln x}{\sqrt{1-x^2}}dx\;,$ Put $x=\cos \phi\;,$ Then $dx = -\sin \phi d\phi$  and Changing Limit, We get  $$\displaystyle I = -\int_{\frac{\pi}{2}}^{0}\cos \phi \cdot \ln(\cos \phi )d\phi = \int_{0}^{\frac{\pi}{2}} \ln(\cos \phi)\cdot \cos \phi d\phi$$  Now Using Integration by parts, We get  $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\frac{\sin^2 \phi}{\cos \phi}d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\frac{(1-\cos^2 \phi)}{\cos \phi}d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\sec \phi d\phi-\int_{0}^{\frac{\pi}{2}}\cos \phi d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\left[\ln\left|\sec \phi+\tan \phi\right|\right]_{0}^{\frac{\pi}{2}}-\left[\sin \phi\right]_{0}^{\frac{\pi}{2}}$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\left[\ln\left|\sec \phi+\tan \phi\right|\right]_{0}^{\frac{\pi}{2}}-1$$  Now How can I solve after that, Help Required, Thanks
How is the notation $X\sim N(\mu,\sigma^2)$ read? Is it $X$ follows a normal distribution? Or $X$ is a normal distribution? Or perhaps $X$ is approximately normal..  What if there are several variables that follow (or whatever the words is) the same distribution? How is it written? Move post to different user In the 3d-view workspace, we can use a random color to give our separated object a distinct look.  What if I don't like the result and I want to randomize the color again?   Swapping the setting doesn't change the random seed in color.    (Blender 2.80.64.) We regularly get questions about wormholes on this site. See for example  and . Various wormhole solutions are known, of which my favourite is  because it's closest to what every schoolboy (including myself many decades ago) thinks of as the archetypal wormhole.  The trouble is that Visser has pulled the same trick as  of starting with the required (local) geometry and working out what stress-energy tensor is required to create it. So Visser can tell us that if we arrange exotic string along the edges of a cube the spacetime geometry will locally look like a wormhole, but we know nothing about what two regions of spacetime are connected.  My question is this: suppose I construct a Visser wormhole by starting in Minkowksi spacetime with arbitrarily low densities of exotic matter and gradually assembling them into the edges of a cube, how would the spacetime curvature evolve as I did so?  I'm guessing that I would end up with something like Wheeler's bag of gold spacetime. So even though I would locally have something that looked like a wormhole it wouldn't lead anywhere interesting - just to the inside of the bag. I'm also guessing that my question has no answer because it's too hard to do any remotely rigorous calculation. Still, if anyone does know of such calculations or can point me to references I would be most interested. Brace expansion not working in a script TeXLive Pretest 2018: beamer and subfig collide \documentclass{beamer}  \usepackage{subfig}  \begin{document}  test  \end{document}   returns undefined control sequence \@@magyar@captionfix  (It was not so in January; I am on fully updated pretest) How can I get rid of this black hole in my minecraft world? I have interesting bug that I have experienced before but has gone away on its own. This one is not going away though.  I have a giant stripe or chunks missing from my desert biome, I would say probably 7 chunks long.  I'm guessing the chunks are corrupted or something, what was hoping to figure is whether it can repaired or reverted.  Here is a picture of the problem. I think that  is pretty good prima-facia evidence that  is just facilitating the deposition of crap. Last year I hired someone to do some work for our website but accidentally lost the PSD files since then. It's a fairly basic concept but I am fairly inexperienced with using masks.  How can I create the black circular gradient area as shown in this image below? I need to get as close as possible where it matches existing images on the site, why I am hoping its not too hard.  Thank you! How can I unlock my phone? I locked it using Android Device Manager, but it claims I entered a wrong password I have an Samsung Galaxy S3 Neo.  Bus 002 Device 042: ID 04e8:6860 Samsung Electronics Co., Ltd GT-I9100 Phone [Galaxy S II], GT-I9300 Phone [Galaxy S III], GT-P7500 [Galaxy Tab 10.1]   The keyboard &amp; language setting is português.  I today installed Android Device Manager and to test it's functionality I locked the phone with the password: "abcd" (4 letters were required).  After entering the password and confirming with the "OK" button of my keyboard, the phone showed me that the password is wrong. "Senha incorreta".  I reset the password after some tries to "123456" and the phone reloaded the lock screen. Again the password showed up to be wrong. I also tried the alphabetic password: "abcdefg" - same error.  I changed the language of my Google account from German to Portuguese and reset the password to "123456" again and I tried to really reboot my phone and unlock the SIM card again. But anyways I can not get through my lock of the Android Device Manager.  I tried to enter the password 35 times wrong with hope to get a chance to authenticate with my Google Account login.  I also tried to connect the phone with adb but it does not show up in this state with:  adb devices   So what can I do to get access to my phone again? I am interested in good integer approximation from below and from above for binary Log(N!).  and  provides only a general idea but not exact values. In other words I need integers A and B so that  A &lt;= Log(N!) &lt;= B Why does parameter expansion with spaces without quotes work inside double brackets "[[" but not inside single brackets "["? I'm confused with using single or double brackets. Look at this code:  dir="/home/mazimi/VirtualBox VMs"  if [[ -d ${dir} ]]; then     echo "yep" fi   It works perfectly although the string contains a space. But when I change it to single bracket:  dir="/home/mazimi/VirtualBox VMs"  if [ -d ${dir} ]; then     echo "yep" fi   It says:  ./script.sh: line 5: [: /home/mazimi/VirtualBox: binary operator expected   When I change it to:  dir="/home/mazimi/VirtualBox VMs"  if [ -d "${dir}" ]; then     echo "yep" fi   It works fine. Can someone explain what is happening? When should I assign double quotes around variables like "${var}" to prevent problems caused by spaces? What is the difference between \emph{...} and {\em ...}?  When to use each of them? Do we actually get more sick (flu/cold) during winter? The word flu derives from the Italian phrase "influenza de freddo" meaning "influence of the cold".  Indeed it is that time of the year when my colleagues seem to have the flu/cold more often than not. I remember during university, winter lectures would be riddled with people coughing and sneezing - the so called "freshers flu"  Questions   Is there statistical evidence for this claim? If so, what are the causes/factors involved?   asks why our noses get runnier during cold whether, is this related? I'm going to guess that the answer is "no" based on the below error message (and ), but is there anyway to perform a cross-database query using PostgreSQL?  databaseA=# select * from databaseB.public.someTableName; ERROR:  cross-database references are not implemented:  "databaseB.public.someTableName"   I'm working with some data that is partitioned across two databases although data is really shared between the two (userid columns in one database come from the users table in the other database). I have no idea why these are two separate databases instead of schema, but c'est la vie... How to use GeoPandas buffer function to get buffer zones in kilometers? Simulation of interacting particles I am looking for an option to require a passcode to power off my iPhone. That way, if the phone is lost, the person who finds it cannot simply turn it off, rendering Find My iPhone useless. Is there a way to accomplish this? The Safari Back Button Problem I do some minor programming and web work for a local community college. Work that includes maintaining a very large and soul destroying website that consists of a hodge podge of VBScript, javascript, Dreamweaver generated cruft and a collection of add-ons that various conmen have convinced them to buy over the years.   A few days ago I got a call "The website is locking up for people using Safari!" Okay, step one download Safari(v3.1.2), step two surf to the site. Everything appears to work fine.  Long story short I finally isolated the problem and it relates to Safari's back button. The website uses a fancy-pants javascript menu that works in every browser I've tried including Safari, the first time around. But in Safari if you follow a link off the page and then hit the back button the menu no longer works.  I made a pared down webpage to illustrate the principle.  &lt;!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"&gt; &lt;html&gt; &lt;head&gt;&lt;title&gt;Safari Back Button Test&lt;/title&gt; &lt;meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"&gt; &lt;/head&gt; &lt;body onload="alert('Hello');"&gt; &lt;a href="http://www.codinghorror.com"&gt;Coding Horror&lt;/a&gt; &lt;/body&gt; &lt;/html&gt;   Load the page and you see the alert box. Then follow the link off the page and hit the back button. In IE and Firefox you see the alert box again, in Safari you do not.  After a vigorous googling I've discovered others with similar problems but no really satisfactory answers. So my question is how can I make my pages work the same way in Safari after the user hits the back button as they do in other browsers?  If this is a stupid question please be gentle, javascript is somewhat new to me. Javascript to check whether a checkbox is being checked or unchecked