Evaluate $\int_0^1 \frac{\ln(1-x^2)\arcsin^2 x}{x^2} {\rm d}x$
What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$?
Why is $2\pi i \neq 0?$ What are the basics needed to learn quantum computing? I was very inspired by Michio Kaku's  on the possibilities of quantum computing and also listening to Talia Gershon's  on it.   As I come from a business &amp; analytics background, what materials can I begin exploring to prepare to learn more about the field of quantum computing? Eg. what would I need to study to understand Peter Shor's algorithm,  on physics, etc. Is there any problem if the last name prints first than the first name on an air ticket? Is there any problem if the last name prints first then the first name in international air ticket? Semigroup analogue to the classification of finite simple groups? Is there a semigroup analogue to the classification of finite simple groups?  If so what are some of the major results?  --Edit, reference links--- Unique (non-repeating) random numbers in O(1)? I'd like to generate unique random numbers between 0 and 1000 that never repeat (i.e. 6 doesn't show up twice), but that doesn't resort to something like an O(N) search of previous values to do it. Is this possible? Bad Eyesight and Evolution Why do many humans have bad eyesight, such as near-sightedness, which hampers performance in a wide variety of tasks? Shouldn't there be evolutionary pressure towards better eyesight? How do I install drivers for the Atheros AR8161 Ethernet controller? How to design a command inquiring internally required packages? Escape character in LaTeX I need to output the below text but since \ is a special character, I cannot:  [RegularExpression(@"\d+")]   Also sometimes I need use dollar sign $ as well but it seems to be a special char, too.  What is the way to escape those in LaTeX?  UPDATE:  I used \verb as suggested but I am unable to run commands inside it. Also, in the output the font and the opacity of the text is different than the default one:  \verb|[RegularExpression(\newline @"\d+")]| In particular, when $x=y=z$ isn't the above expression equal to 1 and not -1? Can there be a function that's even and odd at the same time? Infinite algebraic extension of a finite field I have recently started studying algebraic field extensions and I got to know that algebraic closures $\overline{F}$ of finite fields $F$ are infinite. Therefore, I've asked myself the following question and couldn't come up with an answer.  Suppose $F$ was a finite field, i.e. $|F| &lt; \infty$. Is it possible to construct an algebraic extension $F'$ of $F$ sucht that $|F'| = \infty$ and $F' \subsetneq \overline{F}$? Meaning, that $F'$ is not algebraically closed. The classical book on commutative algebra Introduction to Commutative Algebra, by Atiyah and Macdonald, has the following as exercise I.11.       A ring is Boolean if $x^2=x$ for any $x$ of $A$. In a Boolean ring $A$, show that     i) $2x=0$ for all $x\in A$;    ii) Every prime ideal of $A$ is maximal, and its residue field consists of two elements;   iii) Every finitely generated ideal of $A$ is principal.     I can but solve the first two of them; for the last one, I cannot even perceive the situation here, for it will not, generally speaking, be a Dedekind domain, whereof my knowledge is somewhat more. I have hitherto tried to find one element for each ideal of two generators which can stand the stead of them, and then proceed by induction. But even the first step fails to produce any result. Thanks for any hints and inspirations in advance. How to start Ubuntu in Console mode I want to know if there is a way to switch to console mode from the boot menu.  I have an NVIDIA 1070 and every time there is an update, the GUI stops working. All I need to do is re-install the drivers; however, to do that I need to be in console mode. EDIT: I should have added more information to this question but I'll comment on all answers. In short: I've tried text instead of quiet splash I do see booting activity in text mode; however, I don't get a login screen and ctrl + Alt + F1 doesn't help because GDM keeps trying to restart. Previously I managed to kill GDM through multiple tries but is not working since the latest update. EDIT: After searching for runlevel, I've found  but I think it required me to have a running system to make changes -- please correct me if I'm wrong. Prove that $1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$ Prove that $$1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$$   $\bf{My\; Try::}$ Using Binomial expansion of  $$(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2}x^2+\frac{n(n+1)(n+2)}{6}x^3+.......$$  So we get $$nx=\frac{1}{3}$$ and $$\frac{nx(nx+x)}{2}=\frac{1}{3}\cdot \frac{3}{6}$$  We get $$\frac{1}{3}\left(\frac{1}{3}+x\right)=\frac{1}{3}\Rightarrow x=\frac{2}{3}$$  So we get $$n=\frac{1}{2}$$  So our series sum is $$(1-x)^{-n} = \left(1-\frac{2}{3}\right)^{-\frac{1}{2}} = \sqrt{3}$$  Although I know that this is the simplest proof, can we solve it any other way someting like defining $a_{n}$ and then use Telescopic sum.  Thanks. The RSA signature operation is basically the same as encrypting with the private key.  In particular, both operations use the same kind of keys.  Is it safe to use the same RSA keypair both for encryption / decryption and for signing / verification? Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model How can two dates be compared in a shell?  Here is an example of how I would like to use this, though it does not work as-is:  todate=2013-07-18 cond=2013-07-15  if [ $todate -ge $cond ]; then     break fi                              How can I achieve the desired result? How to access "The grimm Troupe" and "Hidden dreams" content in Hollow Knight? In these two official announcements from Team Cherry, it's said:       Grimm’s quest, and all other new features can be accessed at any point   in your Hollow Knight journey.        Hollow Knight: Hidden Dreams content is accessible at any point in the   game, whether starting fresh or scouring for completion.   I still haven't find any way to start the quests from the menus or during beginning of the game. Some sources say the content can only be accessed late in the game from specific locations.  So, is Team Cherry simply lying in our faces? "Welcome to emergency mode!" Think it is a fsck problem My computer booted to a black screen with this error message.  Welcome to emergency mode! After logging in,type "journalctl -xb" to view    system logs, "systemctl reboot" to reboot, "systemctl default" or ^D to    try again to boot into default mode.   journalctl -xb snippet (what I think is wrong):  -- Unit systemd-fsckd.service has begun starting up. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1 contains a file system with errors, check forced. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1: Inodes that were part of a corrupted orphan linked list found. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: (i.e., without -a or -p options) juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: fsck failed with error code 4. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: Running request emergency.target/start/replace juli 09 15:40:16 kim-SSD-Sationary systemd[1]: systemd-fsck-root.service: main process exited, code=exited, status=1/FAILURE juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Failed to start File System Check on Root Device. -- Subject: Unit systemd-fsck-root.service has failed -- Defined-By: systemd -- Support: http://lists.freedesktop.org/mailman/listinfo/systemd-devel --  -- Unit systemd-fsck-root.service has failed. --  -- The result is failed. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Unit systemd-fsck-root.service entered failed state. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: systemd-fsck-root.service failed. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Starting Remount Root and Kernel File Systems... -- Subject: Unit systemd-remount-fs.service has begun start-up -- Defined-By: systemd -- Support: http://lists.freedesktop.org/mailman/listinfo/systemd-devel   I ignored other errors like: ACPI PCC probe failed, nvidia not proprietary, etc.  I can access my PC by pressing Ctrl+D , but it is annoying.