The integral to be evaluated is  $$\int \frac {x^7-1}{\log x}dx$$  The actual question was to find the definite integral of the above in the limits $0$ to $1$. I could not apply any of the usual tricks to find that integral, hence I thought finding the indefinite integral first and then applying the limits would be the best way to evaluate the same.  I tried applying trigonometric substitution, which did'nt work. I also tried putting $x=e^t$, hoping to simplify the integral but that did not give me any results either.   How should I proceed to evaluate this integral?
/A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand?
I'm in UV Editor and trying to snap the 2d cursor to the vertex ( while the vertices are visible). I turn ON snap and set it to 'vertex mode' , but the 2d cursor won't snap to any visible vertex at all. I can only make it snap to the grid (increment node). Is it a bug ? or did i miss something here ? i'm using 2.8. I was recently told (  ):     You cannot delete a class in production directly. You will need delete   the class from your sandbox and then deploy the deletions to your   production org.   When you deploy from sandbox to production, the   missing classes will come up in red and you can choose to deploy these   deletions to Production.   However, I'd like some clarification because finding information on how to do this is like pulling teeth, unfortunately.  Am I supposed to:   Delete the Trigger in Sandbox Make a deployment package in Sandbox Add the Trigger which is going to be listed in RED now Deploy (and that deletes it from Production)?   I am not seeing deleted Triggers in Red in Sandbox.  Are they NOT listed under "Triggers" for deployment, but elsewhere?   Frustrated = true.   Does this ONLY apply to Classes??  UPDATE:  I just tried again, with a CLASS, and it still doesn't show up at all as a Deployable package (red or not) after I delete it in the Sandbox.  UPDATE 2:  I was just able to delete a class, that I know was written via IDE from a third party, in both Sandbox and Production.  So now I'm wondering if it has to do with where it was originally deployed from (if from IDE, IDE can delete it - it from Sandbox, you have to jump through hoops).  :-( Why can't polynomials have negative exponents or division by a variable Why can't:  $$2x^{-3} - 3x$$  or  $$\frac{1}{2x}$$  be polynomials too?  Why have a definition that excludes these algebraic forms? pgfplots fillbetween with multiple curves I have a TikZ picture like this:  \begin{tikzpicture}   \begin{axis}[     axis lines=middle,     xmin=-0.5,xmax=3.5,ymin=-0.5,ymax=3.5   ]   \addplot[very thick,blue, samples=300, domain=0:3, name path=A] {sqrt(x)};    \addplot[very thick,yellow!70!red, samples=300, domain=0:3, name path=B] {9-3*x};    \addplot[very thick,red!90!teal, samples=300, domain=0:3, name path=C] {3};    \addplot[very thick,green!70!magenta, samples=300, domain=0:3, name path=D] coordinates{(1/9,-6) (1/9,6)};   \end{axis} \end{tikzpicture}     I’d like to fill the area enclosed by these four curves.  Similar questions have been answered using pgfplots’s fillbetween package, but it only seems to support filling between two curves. What are my options? For what $n$ is $U_n$ cyclic? I currently have an ec2 &quot;t2.nano&quot; instance up and running. I would like to switch to a &quot;t2.micro&quot;. In order to do that, I need to stop the instance. But I would lose my IPv4 address in the process. And I do not want that (because I have several ssh tunnels up and running from rasps in the wild that are currently hard coded to connect to this IPv4 address). I decided to create an Elastic IP address. But I understand that by creating an Elastic IP address and assigning it to my current ec2 instance, I would lose my current IPv4 address too. Is it possible to create an Elastic IP address from the current IPv4 address of my ec2 instance? If so, how can I do this operation? A grim and tragic question perhaps, but I'm curious:  What happens when a person that co-signed on a loan dies? Is the co-signer's estate potentially liable until the loan is paid off, the same as if the co-signer had borrowed the money themselves?  Or, does the responsibility for the loan die with the co-signer?  Does the bank/lender take on that risk? I noticed that the reputation from the main site carried over to meta on Gaming (at least the first upvote), I guess that doesn't work the other way around, right?  Is it then impossible to have a meta reputation higher than the site reputation? Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$ Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$.    Challenge: Prove  $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}.$$   I have two proofs, one of which is partially combinatorial.    I'm posing this problem partly because I think some folks on this site would be interested in working on it and partly because I would like to see a purely combinatorial proof.  (But please post any proofs; I would be interested in noncombinatorial ones, too.  I've learned a lot on this site by reading alternative proofs of results I already know.)  I'll wait a few days to give others a chance to respond before posting my proofs.  EDIT: The two proofs in full are now given among the answers. Slipstream Windows 7 + Service Pack 1 Finding the limit of $\left(\frac{n}{n+1}\right)^n$ Showing a compact metric space has a countable dense subset I know that this is separable, but that has not been covered by the text at this point. I believe my proof (below) to be correct, but not very rigorous. Assuming that it actually is correct, could someone please help me make it more formal? (If it's not correct, please help with the proof). Thanks.  Proof: Let $X$ be a compact metric space. Let $B_n = \{B(x,\frac1{n}) : x\in X\}$. be the collection of open balls centered at $x$. Let $\{x\}^{(n)}$ be the set of centers for some particular $n$. Then by compactness of $X$, there are also a finite number of centers for a particular $n$ (corresponding to each open ball).   [This is where I have trouble properly saying what's on my mind:]  Consider $B(x,r_0)$, where $r_0 = \frac{1}{n_0}$. When we decrease the magnitude of $r_0$ to $r_1$, we must still be able to cover $X$. Meaning some open ball(s) must be "formed" where $B(x,r_0)$ was once "covering". So by adding open balls, we also add to our collection of centers. This means at $B(x,r_0)$, there was originally a point $y \in B(x,r_0)$ s.t. $B(y,r') \subset B(x,r_0)$ for some $r'$. That is, $B(x,r_0)$ contained another point of $X$. Thus, by definition, $X$ contains a countable dense subset. Why does instanceof return false for some literals? &quot;foo&quot; instanceof String //=&gt; false &quot;foo&quot; instanceof Object //=&gt; false  true instanceof Boolean //=&gt; false true instanceof Object //=&gt; false false instanceof Boolean //=&gt; false false instanceof Object //=&gt; false  12.21 instanceof Number //=&gt; false /foo/ instanceof RegExp //=&gt; true  // the tests against Object really don't make sense  Array literals and Object literals match... [0,1] instanceof Array //=&gt; true {0:1} instanceof Object //=&gt; true  Why don't all of them? Or, why don't they all not? And, what are they an instance of, then? It's the same in FF3, IE7, Opera, and Chrome. So, at least it's consistent. Measuring accuracy of a logistic regression-based model What is the best time to ask questions? What is a good set of naming conventions to use when developing on the Force.com platform? I've looked in the Apex developer's guide and a saw the   which has basically only has this:   We recommend following Java standards for naming, that is, classes start with a capital letter, methods start with a lowercase verb, and variable names should be meaningful.   I'm looking for something more in depth, such as end all Controllers with Controller and their tests with ControllerTest, etc.  What is a good set of naming conventions to use when developing on the Force.com platofrm?  It would be preferable if the answer has something that handles custom objects, classes, visualforce pages, and components and not just Apex classes. Re-use aliased field in SQL SELECT statement Method overloading in Objective-C? Checking if a worksheet-based checkbox is checked How to auto close LWC quick action from JavaScript?