How to figure out the following integral? I have not been able to solve it from some time. $$\int_0^1\frac{x^{2} -1}{\log x}\,dx$$
I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
Finding all group homomorphisms of $S_n\to\mathbb{C}^*$ and $A_n\to\mathbb{C}^*$ I need to find all group homomorphisms of $S_n\to\mathbb{C}^*$ (for $n\ge2$), and also all group homomorphisms of $A_n\to\mathbb{C}^*$ (for $n\ge2$).  Someone on this site, two years ago, seems to have already asked the same question (). I understand what the questioner says there, but I don't understand it when he says:     Therefore there is exactly one $g:\mathbb{Z}/2\mathbb{Z}\to\mathbb{C}^*$, so all elements of $a\in\mathbb{C}^*$ give us $\operatorname{ord}(a)\mid g[\mathbb{Z}/2\mathbb{Z}]$, so $a$ can only be $-1$ or $1$. Now we know that $f:S_n\to\{-1,1\}$ &hellip;   What is he saying there? And how to go on from there? (I do not understand the answer given either &hellip;)  I hope that someone here can help me further. (And if I'm not allowed to ask this question because is has already been asked two years ago, please tell me what to do instead).  Thank you in advance! Why must I wait two days before accepting my own answer? I got this message today while trying to accept my answer on my own question.  Why must I wait this time, if I am sure that the answer I answered is the right one, just want to understand the point of view?  I faced that just today on Super user and didn't get this message before. Which Windows 7 log file contains device connection/disconnection information? Windows 7 frequently plays the usb/new device connection/disconnection sound. It happens both when the computer is idle and under load. All usb device connections have been checked (though I can't rule out a bad cable yet). I thought a good way to track down the issue would be to wait for the sound and then check the log file for the latest connected/disconnected device. I just need to know which log file has this information.  Edit: I'm going to leave the answer accepted, however the issue persists. I get that sound seemingly randomly throughout the day. Anyone else have any ideas? Installing python module within code I'm Egyptian and flying to Cairo from Tbilisi, via Istanbul. The first flight Tbilisi-Istanbul is on Atlasjet, then from Istanbul to Cairo I am flying on Egyptair. For that, I have to recheck in at Istanbul to my Egyptair flight  Will I need to pass passport control at Istanbul airport to collect my bags and check in on Egyptair? Will I need a transit visa for that? Or it can be all done through neutral zone? Calculate limiting distribution of $\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$ Evaluating $\int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ I need to evaluate $I = \int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ where $a$ is some integer.   I get: $\dfrac{2a(a^2-7)\sin(\pi a)}{a^4 - 10a^2 + 9}$. However $\sin(\pi a)$ is $0$ for all $a$ so $I=0$. But as noted by an answer, there are answers for $a = 1,3,-1,-3$. Clarify or visualize the parameters involved for receiving the Curious badge Is a passport needed when flying within Mexico as a U.S permanent resident? What are good learning resources for a LaTeX beginner? I want to get records from custom object sort by a field as number. The field I want to sort is string type. Is there any way to do that? I am storing result in list, is there a way to sort list based on the string field but sort it as number? Co-signed a loan and I need to have my name removed Creating Ellipse in WGS84, but with metric parameters I have the function  --Ellipse(x,y,rx,ry,rotation,#of segments in 1/4 of ellipse) SELECT ST_Translate( ST_Rotate( ST_Scale( ST_Buffer(ST_Point(0,0), 0.5, $6), $3, $4), $5), $1, $2)   I have points in WGS84 (x,y), radiuses in meters (rx, ry), rotation angle in degrees. And I need a new WGS84 geometry as a result.  As you can see I can't call this function without conversions between WGS84 and some metric SRID. But my points aren't in a definite region. So a SRID which is optimal for one point, throws an error for another point during conversion.  Is there a common way to get an optimal metric projection (SRID) for a WGS84 point? Or at least to use some rough, but world-wide universal metric projection? Or maybe there's a trick for this case? This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. See  for more information.   This is a community wiki for IDEs available on Ubuntu. Please post one IDE per answer (including more than just a screenshot or a link, please at least put a short description).  In your answer, tell us what the IDE is for (which language(s) or if it is RAD capable). postfix doesn't work after iptables set up '₴' is a currency sign used for the Ukrainian Hryvnia. It's in Unicode since 2004. I want to use it in spreadsheets as a sign for money values, but Google Spreadsheets doesn't understand it as a number if I add it after or before my value in the cell. How do I make Google Spreadsheets understand it's currency and display it as a number in my desired format? Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$ Is there a nice presentation for the group $\text{PSL}(2,\mathbb{F}_q)$, for every prime $q$?  (I don't have any particular definition in mind for "nice", other than, say, a small number of generators and relations.) I am aware there are many questions similar to this on MSE but I am having trouble following any of the solutions given.   I have the function given by   $f(x) = \dfrac{xy}{x^2+y^2}\;$ for $(x,y)\neq (0,0)$ and $f(0,0)=0$  I have calculated the partial derivatives and found:   $$\frac{\partial f}{\partial x} = \frac{y(-x^2+y^2)}{(x^2+y^2)^2}, \quad \frac{\partial f}{\partial y} = \frac{x(-y^2+x^2)}{(y^2+x^2)^2}$$  Now I need to show the partial derivatives exist for $(x,y) = (0,0)$.  I am also asked to show it is not continuous at $(x,y)=(0,0)$, based on other answers I've seen it seems like this follows from solving the first part but I fail to see how that follows through as well.   I could just copy the solutions with my function in place as I have seen a lot of answers using the definition of the derivative, but realistically I do want to understand what the thinking behind this is.   Any help would be appreciated. Monos in $\mathsf{Mon}$ are injective homomorphisms. Let $f: M \to N$ be a mono in $\mathsf{Mon}$. I want to prove that the underlying function $U(f): U(M) \to U(N)$ is injective.   How can this be done ? Thanks in advance. Smoke Generator does not work When I try to run the smoke simulator in Blender 2.62 I get no smoke at all. When I attempt to render, the only output is a black screen. What am I doing wrong here that is making the smoke simulation not render?  Is it a mistake I made, or a problem with Blender version 2.62?