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What is the antiderivative of $x^{x^{x^{x...}}}$? What is $\int{x^{x^{x^{x^{\dots }}}}dx}$? I managed to get it down to a complicated series after using $u$-substitution, but I want to see other methods of approach, or if my answer was right.   First I let $$u=x^{x^{x^{x^{\dots }}}}$$, and $du$ would be, in terms of $u$, $$du=  \frac{u}{\frac{1-ln(u)}{u}e^{ln(u)/u}}$$ This makes the integral of $$\int{x^{x^{x^{x^{\dots }}}}dx}=\int \frac{1 - ln(u)}{u}{e^{ln(u)/u} }du$$ Now making $w = ln(u), dw$ becomes $1/u$ and the integral becomes $$\int e^{w/e^{w}}(1-w)dw$$ From there, I distributed and made $2$ integrals of series, using the Taylor polynomial for $e^x$. From there, I integrated and back substituted. I was only trying to find the indefinite integral.   I'm still a junior in high school, so let me know if the answer is beyond my scope of understanding. nginx support for .htaccess / rewrite rules? Differences from Apache? My server is constantly being attacked I am fairly new to the system administration world. I've been working on an application recently and when I check my application server logs, I constantly get various IP addresses trying to ssh into my server by brute force. Here is an example of my server log:   Feb 14 04:07:20 foodwiz3 sshd[1264]: error: Could not load host key: /etc/ssh/ssh_host_ed25519_key Feb 14 04:07:21 foodwiz3 sshd[1264]: reverse mapping checking getaddrinfo for coenamor.columbiansabbatical.com [23.249.167.223] failed - POSSIBLE BREAK-IN ATTEMPT! Feb 14 04:07:21 foodwiz3 sshd[1264]: pam_unix(sshd:auth): authentication failure; logname= uid=0 euid=0 tty=ssh ruser= rhost=23.249.167.223  user=root Feb 14 04:07:23 foodwiz3 sshd[1264]: Failed password for root from 23.249.167.223 port 32997 ssh2 Feb 14 04:07:23 foodwiz3 sshd[1264]: Received disconnect from 23.249.167.223: 11: Bye Bye [preauth] Feb 14 04:13:04 foodwiz3 sshd[1289]: error: Could not load host key: /etc/ssh/ssh_host_ed25519_key Feb 14 04:13:05 foodwiz3 sshd[1289]: reverse mapping checking getaddrinfo for coenamor.columbiansabbatical.com [23.249.167.223] failed - POSSIBLE BREAK-IN ATTEMPT! Feb 14 04:13:05 foodwiz3 sshd[1289]: pam_unix(sshd:auth): authentication failure; logname= uid=0 euid=0 tty=ssh ruser= rhost=23.249.167.223  user=root Feb 14 04:13:07 foodwiz3 sshd[1289]: Failed password for root from 23.249.167.223 port 41562 ssh2   Is this something that is fairly normal, or should I be worried/doing something about it? Debian 10 systemd, how to disable autostart of graphical environments I installed a graphical environment on a server and I have decided that I no longer want to run that environment at boot. How can I disable it with systemd? This is gnome desktop environment. How to prove this sequent using natural deduction? How often is $1+\prod_{k=1}^n p_k$ not prime? How can I go back to the original interface layout? I don't know what the hell I did I just want to get back to modelling Gambler's ruin and coin toss Toolbar options are not displaying in Edit Profile - About me section What (if any) countries are either open to or have set definitive plans to allow international tourism? TikZ: What EXACTLY does the the |- notation for arrows do? I am currently trying to work out how the perpendicular arrows  work, and I fail.  Taking:  \draw [arrow] (node1) -- node[sloped, above] {Caption} (node1 |- node2.north west);   I suspected that the notation meant: Draw an arrow from node 1, to the north-west corner of  node 2. Make the arrow perpendicular to node 1.  Further attempts to draw arrows told me: No, that's not it. I have sadly failed to figure out what exactly the notation is supposed to be doing though! Is there any "explanation for dummies" that will lead me to understand what exactly    node1 |-   Is supposed to do?  Thanks! Finding $\lim_{n\to\infty}\binom{2n}{n}$ I found a limit problem which is $\lim_{n\to\infty}{2n \choose n}$ as we know that $\sum_{i=0}^{n}{n \choose i}^2={2n \choose n}$ now we could transform the given problem as $\lim_{n\to\infty} \sum_{i=0}^{n}{n \choose i}^2$ and as $\sum_{i=0}^{n}{n \choose i}^2$ is a polynomial in $n$, therefore, $\lim_{n\to\infty}\sum_{i=0}^{n}{n \choose i}^2=\infty$ Please tell me if my solution is correct or not,and if it is incorrect then please provide me a solution.  Thanks How do I install Guest Additions in a VirtualBox VM? I've recently installed Ubuntu on Oracle's Virtualbox.  Now I want to install the guest additions.  I started up my Ubuntu guest OS, clicked "Devices" and chose "Install Guest Additions". Nothing happened, and I don't see the CD icon on my desktop.  How can I make this work? I’ve had this issue for months but in the last couple of days it’s got very very bad.  When my phone runs out of battery it needs like 2–3 days to turn back on. At the beginning of this problem the phone would need about 1h. Every time when my phone goes off again the time to switching on goes up.  What does this all look like? Well, when my phone died I put it on the charger, then the logo appears and goes off. When it goes off, a black screen pops out with a little white line on in it, then it turns on back to Apple logo. It keeps doing this until it turns on which is 1 or 2 even now 3 days.  I tried to restore it and also update it but nothing happens, literally nothing happened. Can we get notification in the tab (change of title)? I'm usually running many different StackExchange sites in Firefox using tabs (using ReloadEvery to update every 5 minutes).  Right now to check for activity, I have to manually click on the tab to see if the envelope is lighted.  How about making the icon on the tab i.e. the title text change in some way, denoting that you have activity on the page?  We already add a message count for the chat tabs, so this shouldn't be very complicated.  Possible Example: Example of a finitely generated module with submodules that are not finitely generated I'm looking for an example of a finitely generated module with submodules that are not finitely generated.  I've found  dealing with group (i.e. an example of a finitely generated group with subgroups that are not finitely generated). But I can't figure out whether that question do help to this one.  And I actually want to find a more "module-like" example rather than an example derived from a 'strange' group.  Can you please help? Thank you! Honestly, navigation through Area 51 is a bit confusing and seems to be much different from the other SE sites.    The feature that is very lacking is the footer with all the sites you are subscribed to so that a user can easily return to a different SE site. This one really bugs me.  Can this be easily fixed?  Also, unlike all the other sites there is no "meta" site to add comments specifically about that site.    Again Area 51 seems out of sync with the other SE sites. Why do I get "Required key not available" when install 3rd party kernel modules or after a kernel upgrade? Why do I get "Permission denied (publickey)" when trying to SSH from local Ubuntu to a Amazon EC2 server? I have an instance of an application running in the cloud on Amazon EC2 instance, and I need to connect it from my local Ubuntu. It works fine on one of local ubuntu and also laptop. I got message "Permission denied (publickey)" when trying to access SSH to EC2 on another local Ubuntu. It's so strange to me.  I'm thinking some sort of problems with security settings on the Amazon EC2 which has limited IPs access to one instance or certificate may need to regenerate.  Does anyone know a solution? What is the quantum mechanical explanation of the octet rule? What is the quantum mechanical explanation of the ? In other words, what makes the octet rule be true from a quantum mechanical view? How we explain  what makes some atoms don't follow the octet rule using QM?