I'm attempting to find a closed form for     $$\int_{0}^{1}\frac{\ln\left ( 1-x^{2} \right )\arcsin ^{2}x}{x^{2}}\mathrm{d}x\approx -0.939332$$   I tried to use $$\displaystyle \arcsin^{2}x=\frac{1}{2}\sum_{k=1}^{\infty }\frac{\left ( 2x \right )^{2k}}{k^{2}\dbinom{2k}{k}}$$ but it didn't work and became even more complicated. Any help would be appreciated.
I encountered the integral $$\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx = -0.9393323982...$$ while researching the evaluation of harmonic sums. Mathematica 11 is not able to evaluate this integral, and is not able to evaluate the underlying indefinite integral.  I considered using the Maclaurin series for the expression $\sin ^{-1}(x)^2$ to evaluate this integral. Using this Maclaurin series, it is easily seen that the problem of evaluating the above integral is equivalent to the problem of evaluating the series $$\sum _{n=1}^{\infty } \frac{2^{2 n-1} H_{n-\frac{1}{2}}}{(1-2 n) n^2 \binom{2 n}{n}}=-0.9393323982...$$ which Mathematica 11 is not able to evaluate directly.  I also considered using the Maclaurin series for the expression $\log \left(1-x^2\right)$ to evaluate the above integral. Using this Macluarin series, it is easily seen that this integral is equal to $$\frac{1}{4} \left(16 \pi  C-\pi ^3-\pi ^2 \log (4)\right)+\sum _{n=1}^{\infty } \frac{\, _3F_2\left(\frac{1}{2},\frac{1}{2},1;\frac{3}{2},n+1;1\right)}{n^2-2 n^3}=-0.939332...,$$ but Mathematica is unable to evaluate the above infinite series.  What is a closed-form evaluation of $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$? What techniques can be applied to evaluate this definite integral?
Where can I find a list of all the Stack Exchange sites?  I need to ask a question about cars (where my total knowledge is about zero) but I can't find a cars Stack Exchange site - tried guessing all the obvious (and pun based) names.  I assume the list of a dozen or so at the bottom of this page is only a subset? There must be a site on a topic as geeky as cars!  A list of future/beta sites would also be useful - I found a few on topics I'm interested in only after they had been up for a while. Can I declare a duplicate page via javascript for google? Most sources of documentation state that one can declare a webpage a duplicate of another by applying this tag on the duplicate page:  &lt;link rel="canonical" href="(absolute URL to original page)"&gt;   I could generate the above tag via the script below:  &lt;script&gt; var l=document.createElement('Link'); l.rel='canonical';l.href='(absolute URL to original page)'; document.getElementsByTagName('head')[0].appendChild(l); &lt;/script&gt;   My question is, would google be able to process this javascript and believe I'm declaring the page as a duplicate page? or would I be forced to use the link tag in the header?  I ask because I'm trying to reduce the unnecessary bytes required to be processed before the actual user readable content begins. Installing Broadcom Wireless Drivers Sylvester's determinant identity Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then   $$ \det(I_m+AB) = \det(I_n+BA)$$   where $I_m$ and $I_n$ denote the $m \times m$ and $n \times n$ identity matrices, respectively.  Could you sketch a proof for me, or point to an accessible reference? I have Lubuntu 16.10 installed on my disk mounted as /.  The problem is that the CUPS log file( /var/log/cups/error.log)  grows all the time until I no have free space left on the disk...   So when I delete that file, the space on disk freed up again  I already tried:    fsck disk usage analyzer every apt-cleaning/autocleaning, and dependencies fixing.    How should I proceed?  du -sxh:  9,0G    .  Model: ATA ST3160815AS (scsi) Disk /dev/sda: 160GB Sector size (logical/physical): 512B/512B Partition Table: msdos Disk Flags:   Number  Start   End     Size    Type      File system     Flags  1      32,3kB  50,3GB  50,3GB  primary   ext4            boot  2      50,3GB  160GB   110GB   extended  5      50,3GB  158GB   108GB   logical   ext4  6      158GB   160GB   2136MB  logical   linux-swap(v1)   parted --list &amp;&amp; sudo df -h:  Model: ATA ST3160815AS (scsi) Disk /dev/sda: 160GB Sector size (logical/physical): 512B/512B Partition Table: msdos Disk Flags:   Number  Start   End     Size    Type     File system  Flags  1      32,3kB  50,3GB  50,3GB  primary  ext4         boot  2      50,3GB  160GB   110GB   primary  ext4   Filesystem      Size  Used Avail Use% Mounted on udev            985M     0  985M   0% /dev tmpfs           201M  6,3M  195M   4% /run /dev/sda1        46G   43G  709M  99% / tmpfs          1003M  188K 1003M   1% /dev/shm tmpfs           5,0M  4,0K  5,0M   1% /run/lock tmpfs          1003M     0 1003M   0% /sys/fs/cgroup tmpfs           201M   36K  201M   1% /run/user/1000 /dev/sda2       101G   60M   96G   1% /media/aram/ I've been trying to understand how the second order derivative "formula" works:  $$\lim_{h\to0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$  So, the rate of change of the rate of change for an arbitrary continuous function. It basically feels right, since it samples "the after $x+h$ and the before $x-h$" and the $h^2$ is there (due to the expected /h/h -> /h*h), but I'm having trouble finding the equation on my own.  It's is basically a derivative of a derivative, right? Newtonian notation declares as $f''$ and Leibniz's as $\frac{\partial^2{y}}{\partial{x}^2}$ which dissolves into:  $$(f')'$$ and $$\frac{\partial{}}{\partial{x}}\frac{\partial{f}}{\partial{x}}$$  So, first derivation shows the rate of change of a function's value relative to input. The second derivative shows the rate of change of the actual rate of change, suggesting information relating to how frequenly it changes.  The original one is rather straightforward:  $$\frac{\Delta y}{\Delta x} = \lim_{h\to0} \frac{f(x+h) - f(x)}{x + h - x} = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$  And can easily be shown that $f'(x) = nx^{n-1} + \dots$ is correct for the more forthcoming of polynomial functions. So, my logic suggests that to get the derivative of a derivative, one only needs to send the derivative function as input to finding the new derivative. I'll drop the $\lim_{h\to0}$ for simplicity:  $$f'(x) = \frac{f(x+h) - f(x)}{h}$$  So, the derivative of the derivative should be:  $$f''(x) = \lim_{h\to0} \frac{f'(x+h) - f'(x)}{h}$$  $$f''(x) = \lim_{h\to0} \frac{  \frac{ f(x+2h) - f(x+h)}{h} - \frac{ f(x+h) - f(x)}{h}  }{h}$$  $$f''(x) = \lim_{h\to0} \frac{  \frac{ f(x+2h) - f(x+h) - f(x+h) + f(x)}{h}  }{h}$$  $$f''(x) = \lim_{h\to0} \frac{  f(x+2h) - f(x+h) - f(x+h) + f(x)  }{h^2}$$  $$f''(x) = \lim_{h\to0} \frac{  f(x+2h) - 2f(x+h) + f(x)  }{h^2}$$  What am I doing wrong? Perhaps it is the mess of it all, but I just can't see it. Please help. Successfully connected to a bluetooth speaker, but how do i direct sound there? Is an option missing in the new off-topic reasons? I'm kind of stuck in Terraria. I have full silver armor and 12 hearts (gold/silver gear).  I have just killed the Eye of Cthulhu.  I was thinking about gathering gold armor, but gold is damn scarce. I have tried searching for floating islands (at 406 altitude) but if I go to far I get attacked by Harpies, and I can't build a sky bridge and fight them at the same time.  My meteorite landed in corruption and it's impossible to collect it (I get swarmed by meteor heads with no way to run).  So what is the most natural progression? Should I try and find gold (where?), try to kill the Eater of Worlds, try to mine meteorite or try to fight off imps and collect some hellstone? Suppose a package provides whatever.service file so certain events (including whatever.socket) cause a daemon to start with a certain configuration.  I want to disable this completely. In particular, I want the old socket and any other unit files that might refer to this service by name to be unable to start the daemon.  Instead, I want to create my own unit files for this daemon with a different .socket unit and different environment variables. I also want my override to automatically take account of possible upstream updates to the original unit file (which my current solution of copying whatever.service as whatever-modified.service and editing it unfortunately does not provide).  If I create the following symlinks and files: /etc/systemd/system/whatever.service → /dev/null /etc/systemd/system/whatever-modified.service → /usr/lib/systemd/system/whatever.service /etc/systemd/system/whatever-modified.service.d/fix.conf  systemd considers the new service to be masked as well and refuses to start it. How to avoid this? Can I transfer trophies and data to a new account? Okay I have created a sub account on my ps3 as a new user. I wanted to change my psn online id. So I have copied all game data from the old account to my new account, but when I go to play the game it still says the trophies and data belong to the old account. I don't want to delete the old one and lose all my saved games and trophies, is there a way to either unlock and allow my new user to own the data and trophies or transfer the trophies and data so that my new account may own it? Is there a way to have Cycles skip rendering alpha? If I wanted to create an animation, each frame would have several seconds of rendering a completely transparent part, which adds up obviously. Is there a way to eliminate this waste of time? How to choose between learning algorithms I need to implement a program that will classify records into 2 categories (true/false) based on some training data, and I was wondering at which algorithm/methodology I should be looking at. There seem to be a lot of them to choose from -- Artificial Neural Network, Genetic Algorithm, Machine Learning, Bayesian Optimization etc. etc., and I wasn't sure where to start. So, my questions is:  How should I choose a learning algorithm I should use for my problem?  If this helps, here is the problem I need to solve.    The training data: The training data consists of many rows like this:    Precursor1, Precursor2, Boolean (true/false)   The run I will be given a bunch of precursors. Then,     I choose an algorithm A from different algorithms (or dynamically generate an algorithm), and apply it on every possible combinations of these precursors and collect the "record"s  that are emitted. The "record" consists of several key-value pairs*. I apply some awesome algorithm and classify these records in to 2 categories (true/false). I will generate a table that has the same format as the train data: Precursor1, Precursor2, Boolean   And the whole program is scored based on how many true/false I got right.  *:"Record"s will look like this (hope this makes sense)  Record         [1...*] Score -Precursor1             -Key -Precursor2             -Value   There are only a finite number of possible Keys. Records contain different subset of these keys (some records have key1, key2, key3... other records have key3, key4... etc.).  I actually need 2 learning. One is for step 1. I need to have a module that looks at the Precursor pairs etc. and decide what algorithm to apply in order to emit a record for the comparison. Another is for step 2. I need a module that analyzes the collection of records and categorize them into the 2 categories (true/false).    Thank you in advance! I had a Windows 7 installation on my 1TB SSD laptop with 4 primary partitions. I wanted to install Lubuntu Linux but the process failed with an error which kind of said:     cannot write MBR   It gave me an option to put an MBR somewhere else or to boot without MBR. None of these options worked and it became a useless install. After this failed dual boot attempt, Windows would give a BSOD wit Code 007 ending - translated to inaccessible boot device. That is a huge NTFS partition and I don't think I have any other media to back it up before formatting it.  My only option is somehow rebuilding the old MBR. Hence I started creating a whole bunch of live USB boot environments  including Super GRUB Disk and MiniTool Partition Editor Live CD, but none of this could ever help me recover my lost MBR.  Win PE 4.0 reported the partition as bad disc and the latest version did not even have an option for MBR. I managed to create a live USB with Testdisk Running on Ubuntu. Whatever partition was seen as HPFS on quick  search was written in the MBR and after this the disc is totally unreadable. Earlier I could see my data through Linux and now I can see nothing.  Here are my questions:   When some data could be read (but Windows would not boot) the MBR was shown as dynamic MBR. As far as I know dynamic MBR is a Windows concept and my Linux attempt would not have converted it. While installing Linux I tried to delete 2 primary partitions which I found to be not really useful (I backed up their data first). So I am not sure what created the dynamic MBR.  Did my dual install convert those basic MBR to dynamic MBR? How did that happen? Here is a screenshot of my deep search attempt with TestDisk which is still in progress. I can see 4 NTFS partitions and one of these two system partitions should be the original MBR. When I tried to write both these system partitions (after stopping the deep search momentarily) I got an error like "partition not found". The file system utility showed me that there is no backup sector and that may be the reason why it cannot be read.  What other Resorts do I have, other than doing a plain format and then using some recovery tool to recover all my lost data? I am not even sure I will recover the folder names.  What other options should I be trying to restore my old MBR so Windows can boot?      Update  as of last I managed to create a live USB w/ W7  I used AOMEI Partition assistant 6.0 and on deeper  search I can see the 909 gb partition - it wont let me select it for recovery. I am sorry if my earlier Q was not clear. I referred to restoring an MBR from a partition . What happened earlier  - superficial search revealed all old partitions but NOT this one   and the later partitions showed up on Deep search &amp; I can select Other partitions for recovery but for this one. It wont let me select the partition I want  Here are some more screen shots    So in summary all that happened is   4 primary windows 7 partition -->  wanted to install Linux so I deleted 2 non bootable partitions after backing them up  ended up with BSOD ( inaccessible boot device ) and noticed that some windows partitions showed up as Dynamic MBR  (using Live USB Linux,gparted )  Use standard testdisk procedure to write MBR from the superficial NTFS-HDFS partition.  this is where I screwed up . Hindsight I should have really done a deep search and caught the 901gb partition to write MBR. I am not sure if I can undo this action  after this - life gets worse. I am  on a ventilator ! Now C is unreadable by windows and linux both and testdisk cannot list file ( see screenshots ) and I have been using a series of live USB's with some recovery s/w but no luck. ALL I want is to recreate that 901gb partition and the boot partition and put the MBR on the latter so I can get my windows 7 back again Where does Google Chrome automatically save pages? I opened the following page:   When disconnecting WiFi and even after rebooting the mobile phone, the page can still be read due to being saved automatically.  Left to the URL bar, an indicator looks like this:  This also shows up when opening manually saved pages, which are saved as MHTML files in the default download folder.  However, an automatically saved page can not be re-downloaded manually to get to the download folder, because the arrow is greyed out.I wish to copy some automatically saved pages into the download folder.  Where are the files of automatically saved pages stored? Deriving input based on other parameters in Python script tool? I have a folder and it contains two feature classes, e.g., a.shp and b.shp. The script tool created has two parameters. The first one is the model location, and the second one needs a feature class (a.shp), as shown in the below picture.  What I really want is every time when I set up the model location for the first input, the second input will automatically find the a.shp as the input rather than the user needs to choose a.shp from that folder. I noticed that the dependency is probably what I really need, but there is no code example. Any suggestions?   I am using ArcGIS 10.3.1 Advanced License. Infinite product with sign changes. For infinite products I know the following theorems:  Theorem I. If $a_n \geq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  Theorem II. If $-1 &lt; a_n \leq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  My question is what are necessary and sufficient conditions (possibly including more restrictions than in I and II) for $\prod (1+a_n)$ convergence/divergence as related to  $\sum a_n$ convergence/divergence  if the sequence of terms $a_n$ can switch signs repeatedly but always stay greater than $-1$.  I already know the trivial equivalence of $\prod(1+a_n)$ and $\sum\ln(1+a_n)$. What's wrong with var x = new Array(); Is there a way of creating a filesystem object akin to this:  mknod files p cat file1 file2 ... fileN &gt; files   but such that it can be seeked in, as if it were a regular file? nodejs vs node on ubuntu 12.04